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Motivation and context: For a subset $S$ of a metric space $(M,d)$, the following are two very classical compactness results in Analysis: 1a) The set $S$ is compact if and only if each sequence in $S$...

If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call ...