Compact Accessibility Classes in Topology

• Compact Accessibility Classes are topological categories defined via convergent sequences and compact subset approaches to boundary points in Tychonoff spaces.
• They interpolate classical properties, linking sequential, k’, and Ascoli spaces through refined characterizations using κ-pseudo-open maps.
• These classes inform analyses in function spaces and topological groups by elucidating convergence and compactness under various constructions.

Compact accessibility classes, as introduced in "New classes of compact-type spaces" (Gabriyelyan et al., 24 Oct 2025), constitute a hierarchy of topological space categories that generalize sequential and $k$-type properties by formalizing the manner in which boundary points of open sets in Tychonoff spaces can be attained either by limits of sequences or by compact subsets. These classes are defined with respect to the existence of sequences and relatively compact sets approaching boundary points, interpolate established notions such as $k'$, $\kappa$-Fréchet–Urysohn, Ascoli, and sequential spaces, and facilitate sharper analysis of space properties under standard constructions, quotient mappings, and group topologies.

1. Formal Definitions of the Three Compact Accessibility Classes

The foundational compact accessibility classes are defined for Tychonoff spaces $X$ and open sets $U \subseteq X$ as follows:

1. $\kappa$-Sequential Spaces: $X$ is $\kappa$-sequential if for every open non-closed $U \subseteq X$, there exists $z \in \overline{U} \setminus U$ and a sequence $(x_n) \subseteq U$ with $x_n \to z$.
2. Weakly Open-Compact Attainable Spaces (weakly oca): $X$ is weakly open-compact attainable if for every open non-closed $U \subseteq X$, there exists $z \in \overline{U} \setminus U$ and a compact $K \subseteq X$ such that $z \in K \cap \overline{K \cap U}$.
3. Open-Compact Attainable Spaces (oca): $X$ is open-compact attainable if for every open (possibly closed) $U \subseteq X$, for all $z \in \overline{U}$, there exists a compact $K \subseteq X$ such that $z \in K \cap \overline{K \cap U}$.

Every boundary point of every open set is thus "attainable" either by a sequence converging within $U$ or by a relatively compact subset contained in $U$.

2. Relations to Classical Topological Space Classes

The accessibility classes interpolate between sequential and $k$-type properties, fitting into inclusion diagrams established in Proposition 2.7:

• $k$-space $\Longrightarrow$ $k_{\mathbb R}$-space $\Longrightarrow$ Ascoli $\Longleftarrow$ (oca)
• Sequential $\Longrightarrow$ $s_{\mathbb R}$-space $\Longrightarrow$ $\kappa$-sequential $\Longrightarrow$ (weakly oca)
• $k'$-space $\Longrightarrow$ (oca), $\kappa$-Fréchet–Urysohn $\Longrightarrow$ (oca)

Under angelicity, the accessibility classes coincide with classical properties:

• If every compact subspace of $X$ is Fréchet–Urysohn, then $X$ is open-compact attainable iff $X$ is $\kappa$-Fréchet–Urysohn; $X$ is weakly open-compact attainable iff $X$ is $\kappa$-sequential.

3. Characterizations via $\kappa$-Pseudo-Open Maps

Key characterizations use $\kappa$-pseudo-open and weakly $\kappa$-pseudo-open maps ("dag-maps"):

• $X$ is open-compact attainable iff the inclusion $I:\bigoplus_{K \in K(X)}K \to X$ is $\kappa$-pseudo-open.
• $X$ is weakly open-compact attainable iff $I$ is weakly $\kappa$-pseudo-open.
• $X$ is $\kappa$-sequential iff the inclusion $I:\bigoplus_{S \in \mathcal{S}(X)}S \to X$ of all convergent sequences is weakly $\kappa$-pseudo-open.
• $X$ is $\kappa$-Fréchet–Urysohn iff the same inclusion is $\kappa$-pseudo-open.

For $X$, the following are equivalent (Theorem 4.4):

• $X$ is $\kappa$-Fréchet–Urysohn.
• $\bigoplus_{S \in \mathcal S(X)}S \to X$ is $\kappa$-pseudo-open.
• $X$ is a $\kappa$-pseudo-open image of a metrizable locally compact space.
• $X$ is a $\kappa$-pseudo-open image of a Fréchet–Urysohn space.

4. Permanence Properties Under Topological Constructions

These classes exhibit nuanced behavior under subspaces, products, and quotient maps:

• Subspaces: None of the three classes is hereditary. Exception: hereditary weakly oca implies Fréchet–Urysohn (Theorem 3.13).
• Finite Products: Generally, the three properties are not preserved (various counterexamples). If each factor is first-countable, then arbitrary products are $\kappa$-Fréchet–Urysohn, hence also oca and $\kappa$-sequential.
• Connected Products: If $X$ is connected and $Y$ is $\kappa$-sequential (resp. weakly oca), $X \times Y$ also inherits the respective property.
• Special Products: For pass-connected $X$ and dyadic compactum $K$, $X \times K$ is $\kappa$-sequential.
• Quotients: $\kappa$-pseudo-open images preserve $\kappa$-Fréchet–Urysohn and oca; weakly $\kappa$-pseudo-open images preserve $\kappa$-sequentiality and weakly oca. Quotient groups of $k_{\mathbb R}$-group are weakly oca, $s_{\mathbb R}$-group are $\kappa$-sequential, $k'$-group or $\kappa$-Fréchet–Urysohn group are oca.

5. Examples and Counterexamples

The paper (Gabriyelyan et al., 24 Oct 2025) provides instructive examples:

Space / Construction	Accessibility Class Outcome	Key Properties or Failure
$\varphi = \bigoplus_{n \in \mathbb N}\mathbb R$	Sequential,\ not oca	Sequential; fails open-compact attainability
Arens fan $S_2(\lambda)$ for countable $\lambda$	$\kappa$-sequential,\ not oca	Not oca for uncountable $\lambda$
Non-discrete $P$-space	Sequentially Ascoli,\ not weakly oca	Sequentially Ascoli only
Pseudocompact Ascoli space $X$	Not weakly oca	Ascoli but fails weak oca
$\kappa$-sequential pseudocompact $X$	Not (sequentially) Ascoli	$\kappa$-sequential only
$V(\omega_1) \times V(\omega_1)$	Neither Ascoli nor weakly oca	Fails both properties
$V(\omega_1) \times \mathbb Q$	Neither $\kappa$-sequential nor weakly oca	Fails both properties

These examples demarcate the boundaries and failure modes of the new classes.

6. Applications to Topological Groups and Function Spaces

• Feathered Groups: Every feathered topological group is $\kappa$-Fréchet–Urysohn (Theorem 5.1).
• Bohr Topology on LCA Groups: For locally compact abelian group $G$, the following are equivalent (Theorem 5.3): $G^+$ is $\kappa$-Fréchet–Urysohn, oca, Ascoli, sequentially Ascoli, or a $k'$-space, and $G$ is compact.
• Bohr Topology and $\kappa$-Sequential / Weakly oca: For $G$ locally compact abelian, $G^+$ is $\kappa$-sequential or weakly oca iff $G$ is isomorphic to $\mathbb{R}^n \times H$ with $H$ compact.

These generalizations illuminate new facets of the structure of topological groups, especially in relation to classical Fréchet–Urysohn and $k$-space dualities.

7. Synthesized Accessibility Hierarchy and Structural Significance

The synthesized hierarchy is:

• $\kappa$-sequential $\Leftarrow$ weakly oca $\Leftarrow$ $\kappa$-Fréchet–Urysohn $\Leftarrow$ oca $\Leftarrow$ Ascoli $\Leftarrow$ $k_{\mathbb R}$ $\Leftarrow$ $k$-space.
• Sequential $\Leftarrow$ $s_{\mathbb R}$ $\Leftarrow$ $\kappa$-sequential.
• $k'$ $\Rightarrow$ oca.

These classes provide natural extensions of Arhangel'skii's concepts and new characterizations, particularly via $\kappa$-pseudo-open maps, facilitating advanced topological investigations in function spaces and group topologies. Their introduction sharpens the analysis of compactness, sequentiality, and convergence phenomena, especially beyond the confines of classical dualities.