Compactness and Connectedness in Aura Topological Spaces

This is the second paper in a series on aura topological spaces $(X,\tau ,\mathfrak{a})$, where $\mathfrak{a}:X\to \tau$ 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 ${\tau }_{\mathfrak{a}}$ 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 $({\tau }_{\mathfrak{a}})\times ({\tau }_{\mathfrak{b}})\subseteq {\tau }_{\mathfrak{a}\times \mathfrak{b}}\subseteq {\tau }_X\times {\tau }_Y$ 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.