Let $K_1 \supset K_2 \supset... $ be a sequence of connected compact subsets of $ \Bbb R^2 $. Is $ K = \cap_{i=1}^\infty K_i $ is connected?
55366
Prove that $\bigcap_{k = 1}^\infty C_k$ is also compact and connected.
110485
Nested sequence of non-empty compact subsets - intersection differs from empty set
102484
Connectedness problem and countability of compact subsets of the Sorgenfrey line
65358
Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.
109682
Proposed proof of: If $A \subset B \subset \bar{A}$ and $A$ is connected, then $B$ is connected
148994
$K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
81843
Sequences of sets property
88130
Infinite product $\prod_k (1-z/a_k)(1-z/b_k)^{-1}$ converges uniformly on compacts of$\{\mathbb Im z>0\}$ and is Herglotz
7381
$\sum_{k=1}^{\infty}a_kb_k$ converges for every bounded sequence {$b_k$}, prove that $\sum_{k=1}^{\infty}a_k$ converges absolutely.
153492
Let $p\colon X \to Y$ be a quotient map then if each set $p^{-1}({y})$ is connected, and if $Y$ is connected, then $X$ is connected.
115376
Is $(-\infty,3]\cap(0,1)^c\cap[-1,\infty)$ closed? Open? What are interior points?
29303
Cartesian product of compact sets is compact
58285
Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$
5132
Given $A\cap \overline{B}\neq \emptyset$, prove $A\cup B$ connected.
144383
The closure of a connected set in a topological space is connected
146364
$X$ is connected iff $\forall A\subset X,$ $\partial A\neq\emptyset$
83311
Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}