Let $T_{1}$ and $T_{2}$ be topologies on $X$ such that $(X, T_{1})$ is compact and $(X, T_{2})$ is Hausdorff. Show that $T_{2}$ is not a subset of $T_{1}$.
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See this post – almagest Dec 10 '19 at 19:25
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Does that mean we should assume that $T_{1}$ and $T_{2}$ are not equal so that the result will follow? – Brenda Dec 10 '19 at 19:33
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2Well clearly a space can be both compact and Hausdorff, so if $T_1=T_2$ the result is certainly false (since a set is always a subset of itself). – almagest Dec 10 '19 at 19:34
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Consider $f(x)=x$ from $(X,T_1)$ to $(X,T_2)$.
If $T_1 \subseteq T_2$, $f$ is continuous and thus a homeomorphism (as a continuous map from a compact to a Hausdorff space it will be a closed continuous bijection), and then $T_1=T_2$.
Henno Brandsma
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