Suppose that $A$ is a connected subset of a space $X$ and that $A\subseteq B \subseteq \bar A$. Prove $B$ is connected.
124234
Prove that if $X$ and it's closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected.
30079
Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?
112731
If $Y$ is connected, why is $A\cup Y$ connected in this case?
123149
Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$
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.
116932
Proving a simple graph is a connected graph
137707
How to prove that $(A\cup B) \cap C \subseteq A\cup (B\cap C)$
36044
Show $\Bbb R ^n$ with a finite set removed is still connected.
63442
Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected
102168
Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
144360
For any sets $A$ and $B$, show that $(B\smallsetminus A)\cup A=B \iff A\subseteq B$.
83714
$A \in B$ vs. $A \subset B$ for proofs
144383
The closure of a connected set in a topological space is connected
13415
How to show if $A \subseteq B$, then $A \cap B = A$?
57122
If $E$ is a connected space, then so is $\overline{E}$.
131983
Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$
104411
sum of two connected subset of $\mathbb{R}$