Let X be a space and C a connected subset of X. Suppose C $\subset$ D $\subset$ Cbar . Then D is connected.
109682
Proposed proof of: If $A \subset B \subset \bar{A}$ and $A$ is connected, then $B$ is connected
41047
Subspace Topology of a subset
77621
Continuity of function mapping connected set to connected set
157218
If $E\subseteq X$ is connected, $E\cap A\neq\emptyset$ and $E\cap(X\setminus A)\neq\emptyset$ then $E\cap\partial A\neq\emptyset$
63442
Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected
23225
Union of connected subsets is connected if intersection is nonempty
127173
Closed subset of compact set is compact
135783
Let $T$ and $T^{'}$ be topologies on set $X$ such that $T \leq T^{'}$. If $(X,T^{'})$ is path-connected, is $(X,T)$ path connected?
134073
Definitions of connected space
47810
$C(X)$ is separable when $X$ is compact?
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 $X$
154559
For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
147961
Characterizing simply connected spaces
41080
If A, B, and C are set and $A \subset B ,B \subset C \rightarrow A \subset C$
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.
148994
$K \subset \mathbb{R}^n$ is compact iff it is closed and bounded