Let $X$ and $Y$ be metric spaces, $X$ compact, and $T: X \to Y$ bijective and continuous. Show that $T$ is a homeomorphism.
46970
If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism.
26794
Metrizability of Hausdorff continuous image of compact metric space
117897
A homomorphism induces a continuous map from ${\rm Spec}(A') \to {\rm Spec}(A)$.
151984
Proving the existence of a metric space
145077
A continuous bijection $f:\mathbb{R}\to \mathbb{R}$ is an homeomorphism?
140052
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
94438
Show that the countable product of metric spaces is metrizable
122663
Proving $T$ and $U$ are equal if I know $(X,T)$ and $(X,U)$ are hausdorff and compact
153256
Compact topological space with closed graph implies continuity
34344
Is a function that maps every compact set to a compact set continuous?
103397
Homeomorphisms between infinite-dimensional Banach spaces and their spheres
46788
Compact open topology on space of continuous maps from a discrete space is a product
117502
Proving that a certain continuous function is surjective.
90628
The metric spaces are Tychonoff spaces
153952
A map between metric spaces preserving convergent sequences is continuous
128069
Continuity in Slices + Mapping Compact Sets to Compact Sets = Continuous
99816
Existence of Topologically Transitive Maps on nice Metric Spaces