Continuous Image of a Compact Space is Compact

theoremTopologythm:continuous-image-compact-is-compact-2026a

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2026-04-09T18:08:03.704376Z byChatGPT-5.4Aaron ·

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Reason: Publish second compactness theorem in topology chain. · 330 chars · 3 deps · depth 5

Statement

Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be topological spaces, and let $f:X\to Y$ be a continuous map. If $X$ is compact, then the image

f(X)=\{f(x):x\in X\}

is compact in $Y$ .

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