Prove: $F=\{x_n\}\cup \{x\}$ Is closed
102170
Showing a Sequence and its Limit are Closed
14215
Show that the union $\bigcup A_n$ has empty interior in $X$
46287
Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous, show that $\{x:f(x)\leq g(x)\}$ is closed in $X$
138069
Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
32525
If $f(1)=1$, then is it true that $f(n)=n$ for all $n \in \mathbb{N}\cup\{0\}$.
78117
Closed property of nonempty finite set
74792
$(X,\tau)$ needs to be $T_1$ in order to guarantee that $A'$ is closed?
47434
Proof that $A \subseteq B \Leftrightarrow A \cup B = B$
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
98816
How to prove that $\{f+g<a\}=\bigcup_{r\in{\Bbb Q}}(\{f<r\}\cap\{g<a-r\})$?
66293
$der(\{x\})$ is closed and $x\in der(der(A))$
111538
If $S \subseteq X$ is closed, is $f(S,r)$ necessarily closed?
130299
Prove that if a function $f: X\to Y$ continuous then its graph is closed
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
80876
Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$
12467
Why does this proof work: Closed unit ball in $C_0$ is not compact
137707
How to prove that $(A\cup B) \cap C \subseteq A\cup (B\cap C)$