Compactness and Connectedness in Aura Topological Spaces
Ahu Acikgoz
TL;DR
This paper develops a comprehensive theory of covering and connectivity in aura topological spaces $(X,\tau,\mathfrak{a})$, where $\mathfrak{a}$ assigns to each point an open neighborhood. It introduces five $\mathfrak{a}$-compactness notions, proves a convergence criterion for transitive a-spaces, and shows how $\mathfrak{a}$-compact subsets behave under $\mathfrak{a}$-continuous surjections, with $\mathfrak{a}$-closedness arising in $\mathfrak{a}$-$T_2$ settings. On the connectivity side, the authors define $\mathfrak{a}$-connectedness, $\mathfrak{a}$-path connectedness, $\mathfrak{a}$-components, and $\mathfrak{a}$-local connectedness, establishing fundamental properties and proving that $\mathfrak{a}$-components are $\mathfrak{a}$-closed and often $\mathfrak{a}$-open under local connectedness. The paper also builds subspace and product aura topologies, proves a chain of inclusions for product topologies that collapses to equality under transitivity, and proves a Tychonoff-type theorem for transitive aura spaces. Throughout, all implications are shown to be strict via counterexamples, and the results pave the way for further study of infinite products, paracompactness notions, and categorical aspects of aura spaces.
Abstract
This is the second paper in a series on aura topological spaces $(X, τ, \mathfrak{a})$, where $\mathfrak{a}: X \to τ$ is a scope function with $x \in \mathfrak{a}(x)$. We study covering and connectivity properties in this setting. Five compactness-type notions are defined ($\mathfrak{a}$-compact, $\mathfrak{a}$-Lindelof, countably $\mathfrak{a}$-compact, $\mathfrak{a}$-sequentially compact, $\mathfrak{a}$-limit point compact) and their mutual relationships are determined. For transitive aura functions we obtain a concrete convergence criterion: $(x_n)$ converges to $x$ in $τ_{\mathfrak{a}}$ if and only if $x_n \in \mathfrak{a}(x)$ eventually. We show that $\mathfrak{a}$-compact subsets of $\mathfrak{a}$-$T_2$ spaces are $\mathfrak{a}$-closed and that $\mathfrak{a}$-compactness is preserved under $\mathfrak{a}$-continuous surjections. On the connectivity side, $\mathfrak{a}$-connected, $\mathfrak{a}$-path connected, and $\mathfrak{a}$-locally connected spaces are introduced; $\mathfrak{a}$-components are $\mathfrak{a}$-closed, and they are $\mathfrak{a}$-open when the space is $\mathfrak{a}$-locally connected. We construct subspace and product aura topologies. For products the inclusion chain $(τ_{\mathfrak{a}}) \times (τ_{\mathfrak{b}}) \subseteq τ_{\mathfrak{a} \times \mathfrak{b}} \subseteq τ_X \times τ_Y$ is established, with equality on the left when both scope functions are transitive. A Tychonoff-type theorem for transitive aura spaces is proved. All implications are shown to be strict by counterexamples.