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This question is a follow up to another question of mine, which turned out to be easy (for background on Arhangel'skii's Problem see Arhangel'skii's problem revisited). Recall that a space is linearly …

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 …

Recall that the tightness of a topological space $X$ is defined as the least cardinal $\kappa$ such that for every non-closed subset $A$ of $X$ and every point $x \in \overline{A} \setminus A$, there …

The space $\omega^*$, the remainder of the Cech-Stone compactification of the integers, fails to have all convergence-type properties known to me. Sequentiality. (As a matter of fact $\omega^*$ does …

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 $ …

Is there a ZFC example of a Tychonoff space $X$ such that: $X$ is separable. $X$ has countable tightness (that is, a subset of $X$ is closed if and only if it contains the closure of each one of its …

Alan Dow and Frank Tall recently proved the consistency of the statement Every hereditarily normal manifold of dimension at least two is metrizable. See: Dow, Alan; Tall, Franklin D., Hereditarily nor …

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 …

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 …

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 …

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 …

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 non-empty topological space without isolated points is called maximal if every finer topology on that space has at least an isolated point. The existence of a (Hausdorff) maximal space is a simple c …

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 …

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 …