A topological space $X$ is sequentially compact if every sequence in $X$ has a convergent subsequence.

For example, in a compact topological space, every net has a convergent subnet, and is therefore sequentially compact, as every subnet of a sequence is a subsequence.

When $X$ is a metric space, the following are equivalent:

• $X$ is sequentially compact.
• $X$ is limit point compact.
• $X$ is compact.
• $X$ is totally bounded and complete.