📂Topology

Bolzano-Weierstrass Property and Limit Point Compactness

Definition ¹

If every limit point of every infinite subset of a topological space $X$ belongs to $X$, then $X$ is said to have the Bolzano-Weierstrass property or to be limit point compact.

Theorem

• [1]: Every compact space is a limit point compact space.
• [2]: If $X$ is a metric space, then $X$ being compact is equivalent to it being limit point compact.

Explanation

For example, $[a,b]$ is limit point compact, but $(a,b)$ is not limit point compact. Also, $\mathbb{Q}$ is not limit point compact, since if we consider an infinite subset such as $$ P = \left\{ 3 , 3.1 , 3.14 , 3.141, 3.1415, \cdots \right\} $$ then $\pi \notin P$. Since $\mathbb{R}$ has such subsets, it is naturally not limit point compact either.

What is curious is that, despite the name limit point compact, the definition makes no mention of compactness at all. From the name alone, one might think that limit point compact refers to a special case among compact spaces, but in reality only theorem [1], which is the converse, holds.

Another significance of limit point compactness is that, as in theorem [2], it is useful for showing that some metric space is compact. Since showing that a metric space is compact guarantees the uniform continuity of continuous functions, its usefulness needs no further mention.