Let $T$ and $T'$ be topologies on set $X$ so that $ T \subset T'$. If $(X, T') $ is path connected, is then $(X, T) $ path connected?
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?
88205
How can I define a topology on the empty set?
18039
Let $A$ be a dense subset of $X$ and suppose $A$ is connected in the induced subspace topology, then $X$ is also connected
117877
Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces?
34177
Prove that $\mathbb{R}^2\setminus\mathbb{Q}^2$ is connected without path connectedness
137075
If a topology contains all infinite subsets, then it is the discrete topology
93623
Showing that every connected open set in a locally path connected space is path connected
157830
Definition of locally pathwise connected
73353
Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?
27661
The closed graph theorem in topology
28641
Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, prove that $Y\cup A$ and $Y\cup B$ are connected
58285
Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$
113022
Is the graph $G_f=\{(x,f(x)) \in X \times Y\ : x \in X \}$ a closed subset of $X \times Y$?
133633
A question about path-connected and arcwise-connected spaces
152442
An annulus in $\mathbb R^2$ is path connected
114599
$A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected
115695
If $X$ is a connected subset of a connected space $M$ then the complement of a component of $M \setminus X$ is connected