If two continuous maps into a Hausdorff space agree on a dense subset, they are identically equal
24083
$f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$
19566
Continuous images of compact sets are compact
122663
Proving $T$ and $U$ are equal if I know $(X,T)$ and $(X,U)$ are hausdorff and compact
86782
A $T_4$ space satisfying that 'every continuous map from it to a Hausdorff space is a closed map' is compact.
147331
Compactness and Hausdorffness
156176
Lebesgue measurability of nearly identical sets
85242
Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$
140052
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
105437
How can I show these two metrics give the same topology?
131637
Real projective space is Hausdorff: is this proof correct?
132528
Are metrics that determine the same topology, equivalent?
154978
Some equivalent formulations of compactness of a metric space
100839
If two continuous functions are equal almost everywhere on $[a,b]$, then they are equal everywhere on $[a,b]$
78082
Dense subset of two Banach spaces also dense in the intersection
26794
Metrizability of Hausdorff continuous image of compact metric space
143467
Continuous maps between compact manifolds are homotopic to smooth ones
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space