TITLE: Spaces where a point is always a limit point of any of its neighborhoods
QUESTION [0 upvotes]: Let $ X $ be a topological space. Let $ U $ be an open subset of $ X $, and pick some $ x\in U $.
If $ X = \mathbb R $, then $ x $ is not isolated, but in general this is not the case (any discrete space provides a counterexample).
My question is if there are there any "well-known" and "convenient" requirements on the topology of $ X $ under which a point is always a limit point of any of its neighborhoods. (For example, being Hausdorff alone can't be such a requirement on $ X $, since discrete spaces are Hausdorff in the majority of cases).

REPLY [0 votes]: A space $X$ is called a "crowded space" if it's $T_1$ and has no isolated points.
In a crowded space, all open sets are infinite and every $x$ is a limit point of any of its neighbourhoods, as is easy to see.
(the crowdedness is not a necessary condition, just a common convenient one; an indiscrete space of two or more points also has the property that every point is a limit point of any of its neighbourhoods, trivially).
If $X$ has your property and has two or more points, it cannot have isolated points. A space without isolated points is often called dense-in-itself.
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