Remarks on hD-Topological Spaces

1. Introduction

Levine [12] called a topological space a D-space when every nonempty open subset is dense and proved that this condition is equivalent to hyperconnectedness, i.e., to the requirement that every two nonempty open subsets intersect. Hyperconnected spaces and their mappings were subsequently studied in, among other sources, [2,14]. In the present paper the expression D-space is always used in this sense of Levine; it should not be confused with the unrelated covering-theoretic notion that is also called a D-space in modern general topology.

The notion of an h-open set was introduced by Abbas [1]. Early corrections to some statements and examples were pointed out by Çakallı and İnce Dağcı [3], while İnce Dağcı and Çakallı subsequently emphasized a related topological viewpoint [4]. A decisive structural fact was established by Sharma, Saproo, Billawria and Digra [15]: the family of all h-open subsets of an arbitrary topological space is itself a topology, without any $T_{1/2}$ assumption. We write

$${\tau }_h\left(X\right):=hO\left(X\right)$$

and call $(X,{\tau }_h\left(X\right))$ the associated space.

This associated-topology viewpoint now organizes a series of companion studies. The h-specialization preorder, h-$R_0$ and related difference-set axioms are treated in [10,11]; h-regularity and h-normality are considered in [7]; the pointwise condition h-$R_T$ and the compatibility problem for subspaces and products are studied in [5]; density and countable dense homogeneity are developed in [8]; and further separation properties are considered in [6,9]. The common principle is that any construction expressed solely through h-open and h-closed sets should first be understood as an ordinary construction in $(X,{\tau }_h\left(X\right))$, and only afterwards compared with the original topology $\tau$.

We use throughout the associated-topology, mapping and compatibility results established in the companion papers, together with the classical theory of hyperconnected spaces [2,13,14]. This allows the arguments below to focus on the genuinely new consequences for $hD$-spaces.

The purpose of the present paper is to apply this principle to the h-analogue of Levine’s dense topologies. We call $(X,\tau )$ an $hD$-space when every nonempty h-open subset is h-dense. The central result identifies this exactly with hyperconnectedness of the associated space. Besides clarifying the basic theory, this formulation exposes two points that require particular care. First, the associated topology of a subspace need not be the trace of the ambient associated topology; consequently, a hereditary argument must include an explicit compatibility assumption. Second, the associated topology of a product need not coincide with the product of the associated topologies, so product theorems likewise require compatibility.

We also sharpen the examples and separation picture. A two-point Sierpiński space shows that classical D does not imply $hD$, whereas an infinite Alexandrov chain supplies a genuine nontrivial $hD$ example. In fact, no finite nontrivial example exists: every finite nonempty $hD$-space is a singleton. The same chain shows that $hD$ does not imply h-$R_0$, h-$D_1$ or h-$R_T$, while a two-point indiscrete original topology gives the reverse nonimplications. On the other hand, a product of two infinite cofinite spaces yields a nontrivial $hD$-space whose associated topology is $T_1$. Thus $hD$ is compatible even with h-$T_1$, but not with h-$T_2$ except in the one-point case.

We stress a terminological point. In [10], an $hD$-set abbreviates an h-difference set used in the axioms h-$D_i$. That notion is unrelated to the present term $hD$-space, where the letter D refers to Levine’s dense topologies.

2. Associated Topology and Preliminary Facts

Throughout, $(X,\tau )$ and $(Y,\sigma )$ denote nonempty topological spaces, unless explicitly stated otherwise. We write ${Int}_{\tau }$ and ${Cl}_{\tau }$ for interior and closure in the original topology.

We recall the definition of h-openness introduced by Abbas [1].

Definition 1.

A subset $A\subseteq X$ is calledh-openif

$$A\subseteq {Int}_{\tau }(A\cup U)$$

for every nonempty proper open set $U\in \tau$. The complement of an h-open set is calledh-closed. We write $hO\left(X\right)$ and $hC\left(X\right)$ for the families of all h-open and h-closed subsets of X, respectively.

The fundamental topologicity result for h-open sets was established by Sharma, Saproo, Billawria and Digra [15].

Theorem 1.

For every topological space $(X,\tau )$,

$${\tau }_h\left(X\right):=hO\left(X\right)$$

is a topology on X, and $\tau \subseteq {\tau }_h\left(X\right)$.

The Hausdorff fixed-point property of the associated topology was proved in [8]; we shall use it repeatedly below.

Proposition 1.

(Hausdorff fixed-point property). If $(X,\tau )$ is Hausdorff, then

$${\tau }_h\left(X\right)=\tau .$$

Accordingly, for $A\subseteq X$ we put

$${hCl}_X\left(A\right):={Cl}_{{\tau }_h\left(X\right)}\left(A\right),{hInt}_X\left(A\right):={Int}_{{\tau }_h\left(X\right)}\left(A\right),$$

and

$${hKer}_X\left(A\right):=\bigcap \{U\in {\tau }_h\left(X\right):A\subseteq U\}.$$

When the ambient space is clear, the subscript is omitted. Thus the h-closure is precisely ordinary closure in the associated topology. Since $\tau \subseteq {\tau }_h\left(X\right)$, one has

$$hCl\left(A\right)\subseteq {Cl}_{\tau }\left(A\right)(A\subseteq X).$$

(1)

Definition 2.

For $i\in \{0,1,2\}$, the space $(X,\tau )$ is called anh-$T_i$ spaceif the associated space $(X,{\tau }_h\left(X\right))$ is $T_i$.

This is equivalent to the usual formulations in terms of h-open neighbourhoods used in the companion papers [10,11].

Example 1.

(An h-$T_2$ space which is not $T_1$). Let $X=\{0,1\}$ and $\tau =\{⌀,X\}$. There is no nonempty proper member of τ, so the condition in Definition 1 is vacuous and every subset of X is h-open. Hence

$${\tau }_h\left(X\right)=\mathcal{P}\left(X\right).$$

Thus $(X,\tau )$ is h-$T_2$, whereas the original indiscrete space is not $T_1$.

We shall use the following mapping terminology, consistent with the companion studies [1,7,9].

Definition 3.

A map $f:(X,\tau )\to (Y,\sigma )$ is called

(i)

h-continuousif $f^{-1}\left(V\right)\in {\tau }_h\left(X\right)$ for every $V\in \sigma$;

(ii)

h-irresoluteif $f^{-1}\left(V\right)\in {\tau }_h\left(X\right)$ for every $V\in {\sigma }_h\left(Y\right)$;

(iii)

pre-h-openif $f\left(U\right)\in {\sigma }_h\left(Y\right)$ for every $U\in {\tau }_h\left(X\right)$;

(iv)

anh-homeomorphismif it is bijective and both f and $f^{-1}$ are h-irresolute.

The associated-space interpretation of these mapping notions will be used repeatedly. In particular, h-irresolute maps are exactly the continuous maps between the associated spaces and h-homeomorphisms are exactly their homeomorphisms; these facts are proved in [8]. The corresponding interpretation of pre-h-open maps as open maps between associated spaces is recorded in [9]. Finally, the interpretation of h-continuity as ordinary continuity from $(X,{\tau }_h\left(X\right))$ to $(Y,\sigma )$ is the defining one used in the h-continuity literature; see, for example, [1,7].

3. $hD$-Spaces and Hyperconnectedness

Definition 4.

A topological space $(X,\tau )$ is called an$hD$-spaceif every nonempty h-open set $U\subseteq X$ is h-dense, i.e.

$$hCl\left(U\right)=X.$$

Recall that a topological space is hyperconnected if any two nonempty open subsets intersect. Equivalently, every nonempty open subset is dense; in Levine’s terminology this is exactly a D-space [2,12,14].

Theorem 2

(Associated-topology characterization). For a topological space $(X,\tau )$ the following conditions are equivalent:

(i)

$(X,\tau )$ is an $hD$-space;

(ii)

every nonempty member of ${\tau }_h\left(X\right)$ is dense in $(X,{\tau }_h\left(X\right))$;

(iii)

the associated space $(X,{\tau }_h\left(X\right))$ is a D-space in the sense of Levine;

(iv)

the associated space $(X,{\tau }_h\left(X\right))$ is hyperconnected;

(v)

every pair of nonempty h-open subsets of X has nonempty intersection;

(vi)

there are no two disjoint nonempty h-open subsets of X.

Proof. 

By [8], a subset of X is h-dense in $(X,\tau )$ if and only if it is dense in the associated space $(X,{\tau }_h\left(X\right))$. Hence (i) and (ii) are equivalent. Condition (iii) is simply Levine’s terminology for (ii), while Levine proved that this condition is equivalent to hyperconnectedness [12]; thus (ii)–(iv) are equivalent. Finally, in the associated space the open sets are exactly the h-open sets, so (iv), (v) and (vi) are the standard equivalent formulations of hyperconnectedness; see also [2,14]. □

Theorem 2 also clarifies a useful point that will be used repeatedly: if U is h-open and $hCl\left(U\right)\ne X$, then $X\setminus hCl\left(U\right)$ is a nonempty h-open set disjoint from U.

Corollary 1.

Every $hD$-space is a classical D-space (equivalently, is hyperconnected in its original topology).

Proof. 

Let $U,V\in \tau$ be nonempty. Since $\tau \subseteq {\tau }_h\left(X\right)$ by Theorem 1, both U and V are nonempty h-open subsets of X. By Theorem 2(v), $U\cap V\ne ⌀$. Hence every two nonempty $\tau$-open sets meet, so $(X,\tau )$ is hyperconnected and therefore a D-space in Levine’s sense. □

The converse fails in the smallest possible nontrivial example.

Example 2

(A classical D-space which is not $hD$). Let

$$X=\{0,1\},\tau =\{⌀,\{0\},X\}.$$

The only nonempty proper open set is $\left\{0\right\}$, whose closure is X; hence $(X,\tau )$ is a classical D-space. On the other hand, $\left\{1\right\}$ is h-open, because

$$\left\{1\right\}\cup \left\{0\right\}=X\mathrm{and}{Int}_{\tau }\left(X\right)=X.$$

Thus ${\tau }_h\left(X\right)=\mathcal{P}\left(X\right)$, and the two nonempty h-open sets $\left\{0\right\}$ and $\left\{1\right\}$ are disjoint. Therefore $(X,\tau )$ is not $hD$.

We next give an infinite nontrivial $hD$ example. It will also be used below to separate $hD$ from several h-separation conditions.

Example 3

(An infinite nontrivial $hD$-space). Let $X={\mathbb{N}}_0=\{0,1,2,\dots \}$ and, for $n\in {\mathbb{N}}_0$, put

$$U_n:=\{m\in {\mathbb{N}}_0:m\ge n\}.$$

Consider the Alexandrov topology

$$\tau =\{⌀\}\cup \{U_n:n\in {\mathbb{N}}_0\},U_0=X.$$

Then

$${\tau }_h\left(X\right)=\tau .$$

Every member of τ is h-open because $\tau \subseteq {\tau }_h\left(X\right)$. We prove the reverse inclusion. Let $A\ne ⌀$ be h-open and set $m:=minA$. Fix $n>m$. Since $U_n$ is a nonempty proper τ-open set, h-openness gives

$$m\in A\subseteq {Int}_{\tau }(A\cup U_n).$$

Hence there exists a τ-open neighbourhood of m contained in $A\cup U_n$. Every nonempty τ-open neighbourhood of m is a tail $U_k$ with $k\le m$; therefore, for each $n>m$, we may choose $k_n\le m$ such that

$$U_{k_n}\subseteq A\cup U_n.$$

The finite set $\{0,\dots ,m\}$ contains all $k_n$, so one value k occurs for infinitely many indices $n>m$. The corresponding set of indices is therefore unbounded in ${\mathbb{N}}_0$. Let $r\ge k$ and choose an index $n>r$ for which $k_n=k$. Then $r\in U_k$ and $r\notin U_n$; since $U_k\subseteq A\cup U_n$, it follows that $r\in A$. Thus $U_k\subseteq A$. In particular $k\in A$, and the minimality of m gives $m\le k$. On the other hand $k\le m$, so $k=m$ and therefore $U_m\subseteq A$. Finally, every element of A is at least m, hence $A\subseteq U_m$. Consequently $A=U_m$.

Thus every nonempty h-open set is one of the tails $U_m$, and therefore ${\tau }_h\left(X\right)=\tau$. Any two nonempty tails intersect (indeed $U_m\cap U_n=U_{max\{m,n\}}$), so the associated space is hyperconnected. Theorem 2 now shows that $(X,\tau )$ is an $hD$-space.

The preceding example is necessarily infinite, as the following theorem shows.

Theorem 3

(Finite obstruction). Every finite nonempty $hD$-space is a singleton.

Proof. 

Assume, towards a contradiction, that X is finite, $\left|X\right|\ge 2$, and $(X,\tau )$ is $hD$.

First suppose that $\tau =\{⌀,X\}$. Then there is no nonempty proper $\tau$-open set, so the defining condition for h-openness is vacuous. Hence ${\tau }_h\left(X\right)=\mathcal{P}\left(X\right)$. Since $\left|X\right|\ge 2$, the associated discrete space contains two disjoint nonempty open singletons and is not hyperconnected, contradicting Theorem 2.

Thus $\tau$ is not indiscrete, and there exists a nonempty proper open set. Since X is finite, the family $\tau \setminus \{⌀\}$ has an inclusion-minimal member; fix one and call it U. By Corollary 1, $(X,\tau )$ is hyperconnected. Let $V\in \tau$ be nonempty. Then $U\cap V$ is a nonempty open subset of U, and minimality of U yields $U\cap V=U$. Hence

$$U\subseteq V$$

for every nonempty $V\in \tau$. In particular, if W is any nonempty proper open set, then $U\subseteq W$, so U itself is proper.

Set $A:=X\setminus U$. Since U is proper, $A\ne ⌀$. Let V be any nonempty proper member of $\tau$. From $U\subseteq V$ we obtain

$$A\cup V=(X\setminus U)\cup V=X,$$

and hence

$$A\subseteq {Int}_{\tau }(A\cup V)=X.$$

As this holds for every nonempty proper $V\in \tau$, the set A is h-open. The set U is h-open as well, because every $\tau$-open set is h-open. Thus U and A are disjoint nonempty h-open subsets of X, contradicting Theorem 2(vi). Therefore $\left|X\right|=1$. □

Corollary 2.

If $(X,\tau )$ is Hausdorff, then it is an $hD$-space if and only if X is a singleton.

Proof. 

By Proposition 1, ${\tau }_h\left(X\right)=\tau$ for every Hausdorff space. The assertion is therefore the direct translation of the classical fact that a nonempty Hausdorff hyperconnected space is a singleton; see [2,14]. □

4. Connectedness and Separation Properties

Sharma, Saproo, Billawria and Digra [15] introduced h-connectedness in terms of h-open sets. In the associated topology this is simply ordinary connectedness.

Definition 5.

A space $(X,\tau )$ is calledh-connectedif it cannot be represented as the union of two disjoint nonempty h-open subsets.

Proposition 2.

Every $hD$-space is h-connected, and the converse is false.

Proof. 

The first assertion follows from Theorem 2 and the classical implication “hyperconnected ⇒ connected”; see, for example, [2,14]. It remains only to show that the converse fails.

Let

$$X=\{0,1,2\},\tau =\{⌀,\{0\},\{1\},\{0,1\},X\}.$$

The only subsets of X not already in $\tau$ are $\left\{2\right\}$, $\{0,2\}$ and $\{1,2\}$. The first two are not h-open because, for the nonempty proper open set $U=\left\{0\right\}$,

$${Int}_{\tau }(\left\{2\right\}\cup U)={Int}_{\tau }\left(\{0,2\}\right)=\left\{0\right\},$$

and the defining inclusion fails; similarly, $\{1,2\}$ is not h-open by taking $U=\left\{1\right\}$. Hence ${\tau }_h\left(X\right)=\tau$.

No nonempty proper member of $\tau$ has open complement, so $(X,{\tau }_h\left(X\right))=(X,\tau )$ is connected. Therefore $(X,\tau )$ is h-connected. However, $\left\{0\right\}$ and $\left\{1\right\}$ are disjoint nonempty members of ${\tau }_h\left(X\right)$, so the associated space is not hyperconnected. By Theorem 2, $(X,\tau )$ is not $hD$. □

The Hausdorff level of the associated topology is completely rigid for $hD$-spaces.

Proposition 3.

For a nonempty $hD$-space $(X,\tau )$, the following are equivalent:

(i)

$(X,\tau )$ is h-$T_2$;

(ii)

X is a singleton.

In particular, every nontrivial $hD$-space fails to be h-$T_2$.

Proof. 

By Theorem 2, the associated space $X_h=(X,{\tau }_h\left(X\right))$ is hyperconnected, while h-$T_2$ means precisely that $X_h$ is Hausdorff. A nonempty Hausdorff hyperconnected space is a singleton; this classical fact is included in the mapping theory of hyperconnected spaces, see [2,14]. Hence $\left|X\right|=1$. The converse is immediate. □

The preceding proposition is sharp: the Hausdorff requirement on the associated topology cannot be weakened to $T_1$.

Example 4

(A nontrivial $hD$-space which is h-$T_1$). Let S and T be infinite sets endowed with their cofinite topologies, say α and β, and let

$$X:=S\times T,\tau :=\alpha \times \beta .$$

For $A\subseteq X$, $s\in S$ and $t\in T$, write

$$A_s:=\{v\in T:(s,v)\in A\},A^t:=\{u\in S:(u,t)\in A\}$$

for the horizontal and vertical sections of A. We first determine the associated topology. We claim that

$$A\in {\tau }_h\left(X\right)⟺\begin{array}{c}{A}_s\mathrm{is}\mathrm{cofinite}\mathrm{in}T\mathrm{and}{A}^t\mathrm{is}\mathrm{cofinite}\mathrm{in}S\hfill \\ \mathrm{for}\mathrm{every}(s,t)\in A\hfill \end{array}.$$

(2)

Assume first that A is h-open and let $(s,t)\in A$. The set

$$U:=(S\setminus \{s\left\}\right)\times (T\setminus \{t\left\}\right)$$

is a nonempty proper member of τ. Since $(s,t)\in {Int}_{\tau }(A\cup U)$, there are cofinite sets $P\subseteq S$ and $Q\subseteq T$, with $s\in P$ and $t\in Q$, such that

$$(s,t)\in P\times Q\subseteq A\cup U.$$

For every $v\in Q$, the point $(s,v)$ does not belong to U, and hence $(s,v)\in A$; thus $Q\subseteq A_s$. Likewise $P\subseteq A^t$. Therefore $A_s$ and $A^t$ are cofinite.

Conversely, suppose that the section condition in (2) holds. Let $(s,t)\in A$ and let $U\in \tau$ be nonempty and proper. If $(s,t)\in U$, then the open set U itself is a neighbourhood of $(s,t)$ contained in $A\cup U$, and hence $(s,t)\in {Int}_{\tau }(A\cup U)$. Assume now that $(s,t)\notin U$. Choose a nonempty basic open rectangle $P\times Q\subseteq U$, where $P\subseteq S$ and $Q\subseteq T$ are cofinite. Set

$$P_0:=P\cap A^t,Q_0:=Q\cap A_s.$$

Both $P_0$ and $Q_0$ are cofinite. Since $(s,t)\in A$, one also has $s\in A^t$ and $t\in A_s$. Hence

$$N:=(P_0\cup \left\{s\right\})\times (Q_0\cup \left\{t\right\})$$

is an open neighbourhood of $(s,t)$. Let $(u,v)\in N$. If $u\in P_0$ and $v\in Q_0$, then $(u,v)\in P\times Q\subseteq U$. Otherwise $u=s$ or $v=t$; in the first case $v\in Q_0\cup \left\{t\right\}\subseteq A_s$, while in the second case $u\in P_0\cup \left\{s\right\}\subseteq A^t$. Thus $(u,v)\in A\cup U$ in all cases. Therefore $N\subseteq A\cup U$, so $(s,t)\in {Int}_{\tau }(A\cup U)$. This proves (2).

We now show that $(X,{\tau }_h\left(X\right))$ is hyperconnected. Let $A,B\in {\tau }_h\left(X\right)$ be nonempty, and choose $(s_0,t_0)\in A$ and $(s_1,t_1)\in B$. By (2), the sets $A_{s_0}$ and $B_{s_1}$ are cofinite in T, hence their intersection is nonempty. Choose

$$t\in A_{s_0}\cap B_{s_1}.$$

Then $(s_0,t)\in A$ and $(s_1,t)\in B$, so again by (2) the vertical sections $A^t$ and $B^t$ are cofinite in S. Choose

$$s\in A^t\cap B^t.$$

It follows that $(s,t)\in A\cap B$. Thus any two nonempty h-open sets meet, and Theorem 2 shows that $(X,\tau )$ is an $hD$-space.

Finally, each cofinite factor is $T_1$, and hence the product topology τ is $T_1$. Since $\tau \subseteq {\tau }_h\left(X\right)$, the associated topology is also $T_1$. Thus $(X,\tau )$ is a nontrivial $hD$-space which is h-$T_1$. By Proposition 3, it is not h-$T_2$.

Corollary 3.

The h-$T_2$ hypothesis in Proposition 3 cannot be weakened to h-$T_1$.

Thus h-$T_2$ is precisely the separation threshold in the standard h-$T_i$ hierarchy at which the $hD$ property forces a nonempty space to be a singleton.

We now compare $hD$ with three conditions used in the companion separation theory. Recall that $(X,\tau )$ is h-$R_0$ when $(X,{\tau }_h\left(X\right))$ is $R_0$ [5,11]. An h-difference set is a set $G=U\setminus V$, where $U,V\in {\tau }_h\left(X\right)$ and $U\ne X$; the h-$D_1$ axiom requires two such sets to distinguish each ordered pair of distinct points [10]. Finally, $(X,\tau )$ is h-$R_T$ if, for every $x\in X$, both

$$hKer\left(\right\{x\left\}\right)\setminus hCl\left(\right\{x\left\}\right)\mathrm{and}hCl\left(\right\{x\left\}\right)\setminus hKer\left(\right\{x\left\}\right)$$

have at most one point [5].

Proposition 4.

There is no implication in either direction between $hD$ and any one of h-$R_0$, h-$D_1$ and h-$R_T$. Moreover, $hD$ is compatible with h-$T_1$.

Proof. 

To show that $hD$ implies none of the three properties, we use the tail space of Example 3. There ${\tau }_h\left(X\right)=\tau$. If $m<n$, the h-open set $U_n$ contains n and does not contain m, so the associated space is $T_0$; hence the original space is h-$T_0$.

For $x\in {\mathbb{N}}_0$, a point y belongs to $hCl\left(\right\{x\left\}\right)$ if and only if every tail containing y also contains x. This is equivalent to $y\le x$, and therefore

$$hCl\left(\right\{x\left\}\right)=\{0,1,\dots ,x\}.$$

Likewise, the h-open neighbourhoods of x are exactly $U_0,U_1,\dots ,U_x$, whose intersection is $U_x$; hence

$$hKer\left(\left\{x\right\}\right)=U_x=\{x,x+1,\dots \}.$$

For $x=1$ one has $hCl\left(\right\{1\left\}\right)=\{0,1\}$, whereas the h-open set $U_1$ contains 1 but not 0. Hence $hCl\left(\left\{1\right\}\right)⊈U_1$, so the space is not h-$R_0$. Moreover,

$$hCl\left(\right\{0\left\}\right)=\left\{0\right\},hKer\left(\right\{0\left\}\right)=X,$$

and therefore

$$hKer\left(\right\{0\left\}\right)\setminus hCl\left(\right\{0\left\}\right)=\{1,2,3,\dots \}$$

has more than one point. Thus the space is not h-$R_T$.

It remains to exclude h-$D_1$. Every proper nonempty h-open set has the form $U_n$ with $n\ge 1$. An h-difference set is of the form $G=U\setminus V$ with $U,V\in {\tau }_h\left(X\right)$ and $U\ne X$; therefore every nonempty h-difference set is contained in some $U_n$ with $n\ge 1$ and cannot contain 0. In particular, there is no h-difference set containing 0 but not 1, so the pair $0,1$ cannot satisfy the h-$D_1$ separation requirement. Thus this $hD$-space is neither h-$R_0$, nor h-$D_1$, nor h-$R_T$.

For the converse implications, consider Example 1. Its associated topology is the discrete topology on two points. A discrete space is $R_0$, satisfies the $D_1$ axiom (the two singletons are disjoint difference sets), and satisfies the $R_T$ condition; equivalently, the original space is h-$R_0$, h-$D_1$ and h-$R_T$. It is not $hD$, because its two singleton subsets are disjoint nonempty h-open sets. Hence none of the three properties implies $hD$.

Finally, Example 4 is both $hD$ and h-$T_1$, proving the compatibility assertion. □

Remark 1.

The relations h-$D_0⟺h$-$T_0$ and h-$T_1⟹h$-$D_1$ were established in [11]. Moreover, the singleton closure–kernel characterization

$$hCl\left(\right\{x\left\}\right)=hKer\left(\right\{x\left\}\right)=\left\{x\right\}(x\in X)$$

for h-$T_1$ spaces is [5]; together with [5], it yields the corresponding h-$R_0$ and h-$R_T$ consequences. Thus no proof of these previously established implications is repeated here. Example 4 shows that $hD$ can coexist with all of these weak separation properties. Proposition 4 concerns logical implication, not mutual exclusion. The sharp obstruction occurs at the h-$T_2$ level, by Proposition 3 and Corollary 3.

5. Subspaces and Compatibility

For subspaces, applying the h-construction after passing to the original subspace topology need not agree with taking the trace of the ambient associated topology. Thus an h-open subset of $(Y,\tau {|}_Y)$ need not be h-open in X, even when Y itself is h-open in X. This compatibility issue also occurs in other parts of the associated-topology theory; see [5,9].

We use the subspace-compatibility terminology introduced in the companion studies [5] and [9].

Definition 6.

A subset $Y\subseteq X$ is calledh-subspace compatible in Xif

$${\tau }_h\left(Y\right)={\tau }_h{\left(X\right)|}_Y,$$

where ${\tau }_h\left(Y\right)$ is formed from the original subspace topology ${\tau |}_Y$.

The following counterexample was established in [5]; we recall it because it shows that the compatibility hypothesis used below is genuinely necessary.

Example 5.

(Failure of subspace compatibility). Let

$$X=\{a,b,c\},\tau =\{⌀,\{a\},X\},Y=\{b,c\}.$$

One has

$${\tau }_h{\left(X\right)|}_Y=\{⌀,Y\},{\tau }_h\left(Y\right)=\mathcal{P}\left(Y\right).$$

In particular, Y is h-open in X, but the associated topology of the original subspace does not coincide with the trace of the ambient associated topology. Moreover, $\left\{b\right\}$ is h-open in the original subspace Y but is not h-open in X.

The hereditary statement uses the classical fact that every dense subspace of a hyperconnected space is hyperconnected; see [2,14].

Theorem 4

(Compatible dense heredity). Let $(X,\tau )$ be an $hD$-space and let $Y\subseteq X$ be nonempty. Assume that

$${\tau }_h\left(Y\right)={\tau }_h{\left(X\right)|}_Y$$

and that Y is h-dense in X, i.e., ${hCl}_X\left(Y\right)=X$. Then the original subspace $(Y,\tau {|}_Y)$ is an $hD$-space.

Proof. 

Let $X_h:=(X,{\tau }_h\left(X\right))$. By Theorem 2, $X_h$ is hyperconnected. The compatibility assumption identifies $(Y,{\tau }_h\left(Y\right))$ with the ordinary subspace of $X_h$ carried by Y, while ${hCl}_X\left(Y\right)=X$ says precisely that this subspace is dense in $X_h$. By the classical fact recalled above, $(Y,{\tau }_h\left(Y\right))$ is hyperconnected. Theorem 2 therefore gives that $(Y,\tau {|}_Y)$ is $hD$. □

Corollary 4

(Compatible h-open heredity). Let $(X,\tau )$ be an $hD$-space and let $Y\subseteq X$ be a nonempty h-open, h-subspace compatible subset. Then $(Y,\tau {|}_Y)$ is an $hD$-space.

Proof. 

Since Y is nonempty and h-open in an $hD$-space, Definition 4 gives ${hCl}_X\left(Y\right)=X$. Apply Theorem 4. □

Remark 2.

In an $hD$-space every nonempty h-open subset is automatically h-dense, by Definition 4. Consequently, for a nonempty h-open subset Y, the density hypothesis in Theorem 4 requires no additional verification. The substantive extra assumption in Corollary 4 is therefore precisely the compatibility identity

$${\tau }_h\left(Y\right)={\tau }_h{\left(X\right)|}_Y.$$

Example 5 shows that this identity cannot be omitted merely from h-openness of Y.

6. Mappings

The associated-space characterization yields the following mapping results.

Theorem 5.

(Surjective preservation). Let $f:(X,\tau )\to (Y,\sigma )$ be an h-irresolute surjection. If $(X,\tau )$ is an $hD$-space, then $(Y,\sigma )$ is an $hD$-space.

Proof. 

By [8], h-irresoluteness means that

$$f:(X,{\tau }_h\left(X\right))⟶(Y,{\sigma }_h\left(Y\right))$$

is continuous. It is surjective by hypothesis. The preservation of hyperconnectedness under continuous surjections is classical; see, for example, [2,13,14]. The conclusion now follows immediately from Theorem 2. □

Theorem 6

(Injective reflection). Let $f:(X,\tau )\to (Y,\sigma )$ be a pre-h-open injection. If $(Y,\sigma )$ is an $hD$-space, then $(X,\tau )$ is an $hD$-space.

Proof. 

A pre-h-open map is an open map between the associated spaces. Thus the assertion is the direct associated-space translation of the classical reflection of hyperconnectedness under open injections; see the mapping results in [2,13,14]. Apply Theorem 2 to the domain and codomain associated spaces. □

Corollary 5.

(Invariance). Every h-homeomorphism preserves the $hD$ property.

Proof. 

By [8], an h-homeomorphism is precisely a homeomorphism between the associated spaces. Hyperconnectedness is a topological invariant, so the assertion follows from Theorem 2. □

Corollary 6.

Let $(X,\tau )$ be $hD$ and $(Y,\sigma )$ be h-$T_2$. Every h-irresolute map $f:X\to Y$ has singleton image. In particular, an h-irresolute surjection from an $hD$-space onto an h-$T_2$ space can exist only when Y is a singleton.

Proof. 

By Theorem 2, $(X,{\tau }_h\left(X\right))$ is hyperconnected, and by [8] the map f is continuous from $(X,{\tau }_h\left(X\right))$ to the Hausdorff space $(Y,{\sigma }_h\left(Y\right))$. The classical theorem that every continuous map from a hyperconnected space into a Hausdorff space is constant [2,14] therefore gives the result. □

Corollary 7.

Let $(X,\tau )$ be an $hD$-space and let $(Y,\sigma )$ be Hausdorff. Then every h-continuous map $f:(X,\tau )\to (Y,\sigma )$ is constant.

Proof. 

By Definition 3, h-continuity is ordinary continuity of

$$f:(X,{\tau }_h\left(X\right))⟶(Y,\sigma ).$$

The domain is hyperconnected by Theorem 2. The conclusion is therefore the classical constant-map theorem for Hausdorff targets [2,14]. □

7. Products

For products there are again two topologies that must be distinguished. Let $\{(X_i,{\tau }_i):i\in I\}$ be a nonempty family of nonempty spaces, set

$$X:=\prod _{i\in I}{X}_i,\tau :=\prod _{i\in I}{\tau }_i,$$

and assume that the underlying product set X is nonempty. In general,

$${\tau }_h\left(X\right)\mathrm{and}\prod _{i\in I}{\tau }_h\left(X_i\right)$$

need not coincide; explicit failures are given in the companion studies [5,8,9].

We adopt the product-compatibility terminology of [8].

Definition 7.

The family $\{(X_i,{\tau }_i):i\in I\}$ is calledh-product compatibleif

$${\tau }_h\left(\prod _{i\in I}{X}_i\right)=\prod _{i\in I}{\tau }_h\left(X_i\right),$$

where the associated topology on the left is formed from the original product topology ${\prod }_i{\tau }_i$.

We shall also use the classical product characterization of hyperconnectedness; see, for example, [2,14].

Lemma 1.

(Classical product property). A nonempty product ${\prod }_{i\in I}{Z}_i$ is hyperconnected if and only if every factor $Z_i$ is hyperconnected.

Theorem 7

(Product transfer). Let $\{(X_i,{\tau }_i):i\in I\}$ be an h-product compatible family with nonempty product. Then

$$\left(\prod _{i\in I}{X}_i,\prod _{i\in I}{\tau }_i\right)\mathrm{is}hD$$

if and only if every $(X_i,{\tau }_i)$ is an $hD$-space.

Proof. 

Let

$$X:=\prod _{i\in I}{X}_i$$

with its original product topology ${\prod }_i{\tau }_i$. By h-product compatibility,

$${\tau }_h\left(X\right)=\prod _{i\in I}{\tau }_h\left(X_i\right).$$

Hence the associated space of the original product is exactly the ordinary product

$$(X,{\tau }_h\left(X\right))=\prod _{i\in I}(X_i,{\tau }_h\left(X_i\right)).$$

By Theorem 2, the original product is $hD$ if and only if this associated space is hyperconnected. Since the product is nonempty, Lemma 1 applies and shows that this is equivalent to hyperconnectedness of every associated factor $(X_i,{\tau }_h\left(X_i\right))$. Applying Theorem 2 to each factor gives the desired equivalence with every $(X_i,{\tau }_i)$ being $hD$. □

The h-product compatibility of arbitrary families of Hausdorff spaces was proved in [8]. Combining that result with Theorem 7 gives the following consequence.

Corollary 8

(Hausdorff products). Let $\{(X_i,{\tau }_i):i\in I\}$ be a nonempty family of nonempty Hausdorff spaces with nonempty product. Then

$$\left(\prod _{i\in I}{X}_i,\prod _{i\in I}{\tau }_i\right)\mathrm{is}hD⟺X_i\mathrm{is}\mathrm{a}\sin \mathrm{gleton}\mathrm{for}\mathrm{every}i\in I.$$

Proof. 

By the cited compatibility result, Theorem 7 applies and shows that the product is $hD$ if and only if every factor is $hD$. Corollary 2 then yields the stated singleton condition. □

Remark 3.

The compatibility hypothesis in Theorem 7 is substantive rather than cosmetic. Without it, the associated space of the original product cannot be replaced by the product of the associated factor spaces, and the coordinatewise hyperconnectedness argument does not apply. The finite counterexamples in [5,9] show that the two topologies may already differ for very small factors.

8. Further Consequences

We record two further consequences of the associated-topology characterization.

The next formulation is the standard closed-set characterization of hyperconnectedness, applied to the associated space through Theorem 2; see [2,12,14]. We state it for later reference without repeating the classical proof.

Corollary 9

(Closed-set characterizations). For a topological space $(X,\tau )$ the following conditions are equivalent:

(i)

$(X,\tau )$ is an $hD$-space;

(ii)

every proper h-closed subset of X has empty h-interior;

(iii)

no nonempty h-open subset of X is contained in a proper h-closed subset;

(iv)

the union of any two proper h-closed subsets of X is proper.

Proposition 5.

Let $(X,\tau )$ be a nontrivial $hD$-space. Then the original topology τ is non-indiscrete and has no inclusion-minimal nonempty open set.

Proof. 

Because X is nontrivial, $\left|X\right|>1$. If $\tau$ were indiscrete, there would be no nonempty proper $\tau$-open set, so every subset of X would be h-open. In particular, two distinct singletons would be disjoint nonempty h-open sets, contradicting Theorem 2. Hence $\tau$ is not indiscrete.

Suppose now, towards a contradiction, that U is an inclusion-minimal nonempty member of $\tau$. By Corollary 1, $(X,\tau )$ is hyperconnected. Therefore, for every nonempty open set V, the intersection $U\cap V$ is a nonempty open subset of U. Minimality of U gives

$$U\cap V=U,$$

so $U\subseteq V$ for every nonempty $V\in \tau$.

Since $\tau$ is not indiscrete, choose a nonempty proper open set W. The preceding inclusion gives $U\subseteq W$, and therefore U is itself proper. Put $A:=X\setminus U$, so $A\ne ⌀$. For every nonempty proper open set $V\in \tau$, the inclusion $U\subseteq V$ implies $A\cup V=X$, and hence

$$A\subseteq {Int}_{\tau }(A\cup V)=X.$$

Thus A is h-open. The set U is also h-open because it is $\tau$-open. The two sets U and A are disjoint and nonempty, contradicting the $hD$ property. Hence no inclusion-minimal nonempty open set can exist. □

Example 3 shows that the obstruction in Proposition 5 is compatible with a simple linearly ordered neighbourhood structure: the tails have no minimal nonempty member because

$$U_0⊋U_1⊋U_2⊋\dots .$$

9. Concluding Remarks

The associated topology ${\tau }_h\left(X\right)$ gives the natural and essentially complete framework for $hD$-spaces. The defining statement that every nonempty h-open set is h-dense is exactly Levine’s dense-topology condition in $(X,{\tau }_h\left(X\right))$, and therefore exactly hyperconnectedness of the associated space. This observation both simplifies the original proofs and identifies which parts of the theory are direct transfers from ordinary topology.

The comparison with the original topology nevertheless contains genuine information. Every $hD$-space is a classical D-space, but Example 2 shows that the converse fails. Example 3 supplies a nontrivial infinite $hD$-space, whereas Theorem 3 proves that finite nonempty $hD$-spaces are necessarily singletons. The $hD$ property is strictly stronger than h-connectedness. At the separation level, Proposition 3 gives the exact trivialization threshold: a nonempty $hD$-space is h-$T_2$ if and only if it is a singleton, while Example 4 shows that nontrivial $hD$-spaces may still be h-$T_1$. Proposition 4 further shows that $hD$ is logically incomparable with h-$R_0$, h-$D_1$ and h-$R_T$, even though these properties may coexist with $hD$.

For subspaces and products, the h-construction does not automatically commute with the corresponding ordinary topological operations. Example 5 shows the failure of a naive transitivity assertion for h-openness, while Theorem 4, Corollary 4 and Theorem 7 provide precise transfer statements under the appropriate associated-topology compatibility hypotheses. The previously established automatic product compatibility in the Hausdorff category, recalled before Corollary 8, yields the corresponding $hD$ product characterization. The mapping results follow from the classical behaviour of hyperconnected spaces under continuous images; in particular, Corollary 7 recovers in the present setting the familiar fact that maps of the relevant continuity type into Hausdorff spaces must be constant.