2026/02/07 by Ahu Acikgoz · 1 voice · 1 citation
Mathematics · #math.GN
This is the second paper in a series on aura topological spaces (X, τ, \mathfraka), where \mathfraka: X → τ is a scope function with x ∈ \mathfraka(x). We study covering and connectivity properties in this setting. Five compactness-type notions are defined (\mathfraka-compact, \mathfraka-Lindelof, countably \mathfraka-compact, \mathfraka-sequentially compact, \mathfraka-limit point compact) and their mutual relationships are determined. For transitive aura functions we obtain a concrete convergence criterion: (xn) converges to x in τ_\mathfraka if and only if xn ∈ \mathfraka(x) eventually. We show that \mathfraka-compact subsets of \mathfraka-T2 spaces are \mathfraka-closed and that \mathfraka-compactness is preserved under \mathfraka-continuous surjections. On the connectivity side, \mathfraka-connected, \mathfraka-path connected, and \mathfraka-locally connected spaces are introduced; \mathfraka-components are \mathfraka-closed, and they are \mathfraka-open when the space is \mathfraka-locally connected. We construct subspace and product aura topologies. For products the inclusion chain (τ_\mathfraka) × (τ_\mathfrakb) ⊆ τ_\mathfraka × \mathfrakb ⊆ τX × τY 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.