Let X be a topological space. Show that X is $ T_2 $ if, and only if, $ \alpha $ = {(x, x) $ \in $ X $\times$ X : x $ \in $ X} is closed at ...
2915
$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed
149820
How do you prove the set $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed?
150331
Whenever $f=T(x)$ then $f(x) \geq 0$. Prove $T$ is bounded by closed graph theorem.
152465
Find the intersection of all $T_2$ topologies on an infinite set $X$
62911
Let $(X,\tau)$ be a connected topological space, $Y\subset X$ connected and $U\subset X\setminus Y$ clopen. Prove that $ Y \cup U$ is connected
101467
Let $T:V\to V$ be a linear map, where $V$ is an inner product space. Given $T^2=\frac{T+T^*}{2}$, prove that $T$ is normal and that $T=T^2$.
73353
Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?
16649
Topological space with countable open cover $\{U_\alpha\} $ with each $U_\alpha$ second-countable, is second countable
122190
Kernel of $T$ is closed iff $T$ is continuous
59424
Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
116768
Problem 5, sec. 13 in Munkres' Topology, 2nd ed.: How to prove the assertion if $\mathscr{A}$ is a subbasis?
52629
Example 1, Sec. 22 in Munkres' TOPOLOGY, 2nd edition: How to verify that this map is closed?
10180
How to prove that this set is closed?
31617
Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?
21114
$S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$?
86782
A $T_4$ space satisfying that 'every continuous map from it to a Hausdorff space is a closed map' is compact.
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 $X\times Y$