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

Source abstract

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.