$X$ is closed iff $X$ is complete
88185
Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed
86317
Show that $F\subset Y$ is closed in $Y$ iff $F=Y\;\cap\;H$ where $H\subset X$ is closed in $X$.
27661
The closed graph theorem in topology
81423
Completeness and Topological Equivalence
61930
Intersection of Closed and Compact Set is Closed
157603
Continuity $\iff$ Preimage is closed whenever set is closed
128482
The set of subsequential limits is closed
151923
$F$ closed and connected then...
119105
If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$
99236
Interior of closure of an open set
16735
If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed
122190
Kernel of $T$ is closed iff $T$ is continuous
92588
Showing that $(C_0([0,\infty)), \|\cdot\|_\infty)$ is complete.
55238
Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$
35573
A closed ball in a metric space is a closed set
114608
Proof concerning regular space: there exist a closed set contained in any open set containing $x$
130975
Given a pair of continuous functions from a topological space to an ordered set, how to prove that this set is closed?