Suppose $M⊆\mathbb{R}^n$ is connected and $M⊆N⊆\overline{M}$. Prove that N is connected.
127919
prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected.
114599
$A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected
16654
Unions and Intersections of Open Sets are Open
47680
Prove that the union of the interior of a set and the boundary of the set is the closure of the set
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.
106434
Prove that for a dense set $A$ and an open set $U$, $U\subset \overline{A\cap U}$
93623
Showing that every connected open set in a locally path connected space is path connected
108394
Proof of that every interval is connected
133633
A question about path-connected and arcwise-connected spaces
88301
What are all of the connected subsets of $\mathbb{Q}$?
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
89264
Connectedness in the Quotient Space
143735
Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$
45337
Suppose that $V, W$ are closed in $X$ and that $V\cap W, V\cup W$ are connected. Show that $V$ and $W$ are connected.
44673
$M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time
92740
prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$
39746
How to prove that given set is a connected subset of the space of matrices?