A pair (Y, f) is a compactification of a topological space X iff for every compact space X' and continous map g from X to X', there is a continuous map g' from Y to X' such that g'(f(x))=g(x) for all x in X. Is this true?
2026-03-25 23:35:32.1774481732
Does every compactification have an extension property?
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No. For example, the one point compactification of $(0,1)$ is homeomorphic to the circle $S^1$. Another compactification of $(0,1)$ is $[0,1]$. But there is no continuous map $S^1\to [0,1]$ which extends the map $(0,1)\to S^1$.
The Stone–Čech compactification is a special compactification which satisfies your condition when $X'$ is additionally assumed to be Hausdorff. Moreover, the map $g'$ is unique in this case. This is the universal property of the Stone–Čech compactification.
This raises an interesting question that I don't know the answer to off the top of my head: can other compactifications be characterized in terms of universal properties?