nLab limit point compact space

Definitions

A topological space is limit point compact if every infinite subset has a limit point (in the sense of Definition 2.3 there, i.e., an accumulation point).

Under classical logic, there are several tautologously equivalent ways of formulating the definition. Let us first recall that points $x$ of a closed subset $A$ of a space $X$ are either isolated (meaning that for some neighborhood $U$ of $x$ we have $U \cap A = \{x\}$) or not; in the latter case we say $x$ is a limit point or accumulation point of $A$. More generally, for any subset $A$, a limit point of $A$ is defined to be a limit point of its closure $\bar{A}$.

Thus, a subset $A$ has no limit point iff every point of $\bar{A}$ is isolated, in other words iff $\bar{A}$ equipped with the subspace topology is a discrete space. In view of this, an equivalent definition of limit point compactness is:

• Every closed discrete subspace is finite.

Another characterization focuses in on countability: if there are no infinite closed discrete spaces, then obviously there are no countably infinite closed discrete spaces. Conversely, if there is an infinite closed discrete subspace $A$ of $X$, then because any countably infinite subset of $A$ is also closed and discrete in $X$, we can say that $X$ is not limit point compact iff there exists an infinite closed discrete subspace iff there exists a countably infinite closed discrete subspace. It follows that another characterization of limit point compactness is

• Every countably infinite set has a limit point, or

• Every countable closed discrete subspace is finite.

Limit point compactness is closely related to countable compactness. Indeed, every countably compact space is limit point compact; in the converse direction, every limit point compact space satisfying the $T_1$ separation axiom (namely, that points are closed) is also countably compact. For this reason, limit point compact spaces are also known as “weakly countably compact spaces”. See countably compact space for further details.

Remark

For a metrizable space $X$, the following are equivalent:

• $X$ is a limit point compact space
• $X$ is a countably compact space
• $X$ is a sequentially compact space
• $X$ is a compact space

Terminology

The term “limit point compact” was invented by James Munkres as he was writing his famous text on point-set topology. He writes that other terms that have appeared in the literature are “Bolzano-Weierstrass compactness” and “Fréchet compactness”.

References

• James Munkres, Topology, 2nd edition, Prentice-Hall 1999. ISBN 0-13-181629-2.