Compactness and Connectedness in Aura Topological Spaces

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.

💡 Research Summary

The paper investigates covering and connectivity properties in the framework of aura topological spaces, a recent generalization of classical topological spaces introduced in the authors’ earlier work. An aura space is a triple ((X,\tau,\mathfrak a)) where (\tau) is a topology on (X) and (\mathfrak a:X\to\tau) is a “scope” function satisfying (x\in\mathfrak a(x)) for every point. The function (\mathfrak a) assigns to each point a distinguished open neighbourhood, and from it one derives an auxiliary topology (\tau_{\mathfrak a}\subseteq\tau) consisting of all sets (U) such that (\mathfrak a(x)\subseteq U) for every (x\in U). The authors develop a systematic theory of compactness and connectedness relative to (\tau_{\mathfrak a}).

Compactness. Five notions are introduced:

1. (\mathfrak a)-compactness (compactness in (\tau_{\mathfrak a}));
2. (\mathfrak a)-Lindelöfness (every (\mathfrak a)-open cover has a countable subcover);
3. countably (\mathfrak a)-compact (every countable (\mathfrak a)-open cover admits a finite subcover);
4. (\mathfrak a)-sequential compactness (every sequence has an (\mathfrak a)-convergent subsequence);
5. (\mathfrak a)-limit‑point compactness (every infinite subset possesses an (\mathfrak a)-limit point).

The paper establishes the basic hierarchy \