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A space has a $G_\delta$-diagonal if its diagonal can be written as the intersection of countably many open subsets of the square. A space is submetrizable if it has a weaker metrizable topology. Ever …

In a recent MO post it was noted that Uspenskij's old example of a Tychonoff ccc space with a $G_\delta$ diagonal and arbitrarily large cardinality is not normal. See: How could I see quickly that th …

QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spaces? Recall that a sp …

An open neighbourhood assignment for a topological space $(X, \tau)$ is a map $U: X \to \tau$ such that $x \in U(x)$, for every $x \in X$. A space $X$ is called a $D$-space if for every open neighbour …

Recall that a space is countably compact if every infinite set has an accumulation point. A space is perfect if every closed set is a countable intersection of open sets. One of the classical applicat …

A space $X$ is said to be sequential if whenever $A \subset X$ is not closed then $A$ contains a sequence converging to a point outside of $A$. A space $X$ is said to have countable tightness if for …

Given a topological space $X$, let $X_\delta$ be the topology on $X$ generated by the $G_\delta$ subsets of $X$. Let $c(X)$ be the cellularity of $X$, that is, the supremum of cardinalities of familie …

One way of formulating Arhangel'skii's celebrated theorem about the cardinality of Lindelof first-countable spaces is the following (due to Arhangel'skii and Shapirovskii). For every Hausdorff space $ …

An $L$-space is a hereditarily Lindelof regular space which is not separable. A space is $d$-separable if it contains a dense set which is the countable union of discrete sets. An $L$-space can't …

A space $X$ is called an almost $P$-space if $Int(G) \neq \emptyset$ for every non-empty $G_\delta$ subset $G \subset X$. Every $P$-space (that is, a space where $G_\delta$s are open) is an almost $P …

Background/Motivation A space is scattered if every non-empty subset has an isolated point. A space is pseudoradial if every non-closed set contains a transfinite sequence (a well-ordered net) converg …

An L-space is a regular hereditarily Lindelof space which is not hereditarily separable. Consistent examples of L-spaces are relatively easy to come by (for example, Suslin Lines), but the first const …

Recall that a space is: "Lindelof", if every open cover has a countable subcover. "Linearly Lindelof", if every open cover which is linearly ordered by $\subseteq$ has a countable subcover. "weakly …

A space $X$ is: Lindelof if every open cover for $X$ has a countable subcover. A $P$-space if every $G_\delta$ subset of $X$ is open. Discretely generated if for every non-closed set $A \subset X$ an …

One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is the inter …