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.
12104
How to prove this result involving the quotient maps and connectedness?
146364
$X$ is connected iff $\forall A\subset X,$ $\partial A\neq\emptyset$
28406
composition of certain covering maps
12490
A map is continuous if and only if for every set, the image of closure is contained in the closure of image
149999
fixed point of a continuous map on a projective space
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?
55486
Topological proof to show a complement set of a set is polygonally connected
39746
How to prove that given set is a connected subset of the space of matrices?
15865
Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected.
77621
Continuity of function mapping connected set to connected set
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.
39455
Prove that $(X\times Y)\setminus (A\times B)$ is connected
1453
Prove that the graph is connected
100437
Classification of connected subsets of the real line (up to homeomorphism)
116866
$A\times B$ connected component implies $A,B$ connect components
127919
prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected.
19599
$ d(x_n,y_n) \to d(x,y)$ if $x_n\to x$ and $y_n \to y$