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Does anyone have an intuition about the following: general compact topological spaces might not be sequentially compact, but compact metric spaces are? What is special about metric-induced topologies?

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Spaces that are compact but not sequentially compact: https://topology.pi-base.org/spaces?q=Compact%2B%7ESequentially+Compact

Automatically generated proof that metrizable compact spaces are sequentially compact: https://topology.pi-base.org/spaces?q=Compact%2B%7ESequentially+Compact%2BMetrizable

At the heart of this is a particular result: every countably compact and sequential space is sequentially compact: e.g. theorem 1.20 of https://open.library.ubc.ca/soa/cIRcle/collections/ubctheses/831/items/1.0080490

This applies as every compact space is countably compact, and every metrizable space is first countable and thus sequential.

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Essentially what's special about a metric topology is that it can be described using sequences: e.g. a set $S$ is open iff every sequence converging to a point of $S$ is eventually in $S$.

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