Prove that $\bigcap_{k = 1}^\infty C_k$ is also compact and connected.
26919
Intersection of Connected Sets
94485
Limit point compactness implies compactness.
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
102484
Connectedness problem and countability of compact subsets of the Sorgenfrey line
30770
Prove that $C\left(\bigcup_{\alpha\in I} A_\alpha\right)=\bigcap_{\alpha\in I}C\left(A_\alpha\right)$
72209
show that torus is compact
101712
Let $X$ be obtained by removing finitely many lines from $\mathbb{R}^3$. How do I prove $X$ is connected?
80735
Show that $C_0([a, b], \mathbb{R})$ is not $\sigma$-compact
57943
If $G$ is a convex set, prove that $\mathbb{C}\cup\{\infty\}\setminus G$ is connected.
63529
How to show that this set is compact in $\ell^2$
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) $
137446
Show that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$.
36878
Prove that $\operatorname{SL}(n,\Bbb R)$ is connected.
128710
Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$
140052
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
108394
Proof of that every interval is connected
107701
Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$