Suppose $X$ has two topologies $(X,T),(X,T')$ and both are compact and hausdorffs. Suppose $T \subset T'$. Prove that $T=T'$
100121
Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.
141642
Closure of a certain subset in a compact topological group
16512
Product of $T_1$ spaces is $T_1$
147237
If $A$ and $B$ are compact subsets of $\mathbb R$, then so is $\frac{A}B$.
13132
If the union of two sets is contained in the intersection, then one is contained in the other ($[A\cup B\subseteq A\cap B]\implies A \subseteq B$)
114123
Do the topologies with the same convergent sequences coincide?
83166
Homeomorphism(topological spaces) version of Cantor–Bernstein–Schroeder theorem
37654
Topology: Example of a compact set but its closure not compact
29303
Cartesian product of compact sets is compact
134859
Let $(X, \mathcal{T})$ be a metrizable space. Prove that $X$ Lindelöf implies that $X$ is separable.
41477
Prove $f(S \cap T) \subseteq f(S) \cap f(T)$
66180
Homomorphism between compact Hausdorff spaces.
28641
Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, prove that $Y\cup A$ and $Y\cup B$ are connected
108864
Showing that two topologies give rise to the same topology
110744
How to prove that a compact set in a Hausdorff topological space is closed?
106298
Prove that some topology is not metrizable
147331
Compactness and Hausdorffness