Boundary-Approach Density of Compactification Remainders, Products, Order and Spectra

Let \(X\) be a Tychonoff space and let \(bX\) be a Hausdorff compactification of \(X\). We study the boundary-approach density \(d_b^{\partial}(X)\), an embedding-sensitive cardinal measuring how small a single subset of \(X\) can be while its closure reaches the whole remainder. We examine its behavior under products and as the compactification varies. For a non-empty family \((X_i)_{i\in I}\) of non-empty Tychonoff spaces with Hausdorff compactifications \(b_iX_i\), put \(P=\prod_{i\in I}X_i\) and \(K=\prod_{i\in I}b_iX_i\). We obtain an exact trichotomy. If all factors are compact, the value is \(0\). If exactly one factor is non-compact, the value is the maximum of the boundary-approach density of that factor and the density of the remaining compact product. If at least two factors are non-compact, the value is exactly \(\text{dens}(P)\), independently of the chosen factor compactifications. For a non-compact locally compact Tychonoff space \(X\), the one-point and Stone–Čech compactifications attain the minimum and maximum of the boundary-approach spectrum. If \(X\) also admits a clopen decomposition \(X=\coprod_{\xi<\tau}G_\xi\), where \(\tau\) is infinite and each \(G_\xi\) is non-empty and non-\(\omega\)-bounded, then every infinite cardinal \(\lambda\leq\tau\) is realized by a compactification \(b_\lambda X\), and the compactifications may be chosen to form a chain in the compactification order. When \(\tau=\text{dens}(X)\), the spectrum is the full interval \(\{\lambda:\omega\leq\lambda\leq\text{dens}(X)\}\). In particular, every non-compact locally compact metrizable space has this full spectrum. We also obtain an exact closed-core formula, an intrinsic Stone–Čech characterization, the identity \(d_\beta^{\partial}(X)=\text{dens}(X)\) for non-compact metrizable \(X\), and monotonicity of \(d_\beta^{\partial}\) under continuous maps with dense image.