Let $E$ be a subset of $\mathbb{R}^k$. Show that $E$ is compact if and only if every sequence in $E$ has a converging subsequence to a point in $E$.
32534
Prove that a set $S \subseteq \mathbb{R}$ is closed and bounded if and only if every sequence in $S$ has a sub sequence converging to a point of $S$.
43376
If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved
127173
Closed subset of compact set is compact
55337
Find a sequence of functions $\{g_n\}$ in $C([0,1])$ so that $\overline{\{g_n\}}$ is compact, but $g_n$ does not converge uniformly.
144693
Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$
153573
Counter Example for the union of any number of compact sets is compact in T2-spaces
22854
Let $A , B\subseteq\mathbb{R}$. If $A$ is closed and $B$ is compact, is $A\cdot B$ closed?
32777
Prove: If a sequence converges, then every subsequence converges to the same limit.
116542
Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$[NBHM_PhD Screening Test-2013, Topology]
95993
Isn't every subset of a compact space compact?
106534
"Closed subsets of compact space are compact" with an alternative definition of covering
28391
Prove the compactness of a set B
64694
$C_{c}(X)$ is complete. then implies that $X$ is compact.
8015
Compact set plus a closed set is a closed set
74311
Proving that the Intersection of any Collection of Compact Sets is Compact
86309
Whether the set $A$ is a compact subset of $M_3(\Bbb R)$.
35731
Prove that if $A$ is nonsingular, then the sequence $X_{k+1}=X_k+X_k(I-AX_k)$ converges to $A^{-1}$ if and only if $ρ(I-X_0A)<1$.