$(\mathbb{R}^n,d)$ metric space, $U \subseteq \mathbb{R}^n$ connected and open subset. Show that $U$ is also path-connected
93623
Showing that every connected open set in a locally path connected space is path connected
104411
sum of two connected subset of $\mathbb{R}$
31266
Proof of "the continuous image of a connected set is connected"
42266
Show that $d(x,y)=\min \{1,|x-y|\}$ is a metric on $\mathbb R$
67088
Closure and pathwise connected
14345
If $A\subset\mathbb{R^2}$ is countable, is $\mathbb{R^2}\setminus A$ 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?
132641
Proving that Euclidean Space is a complete metric space
2575
Showing that $\mathbb{R}$ is connected
36878
Prove that $\operatorname{SL}(n,\Bbb R)$ is connected.
108052
Does any connected metrizable space with more than one element have uncountably many open sets, each generating a connected subspace?
13848
Showing that if a subset of a complete metric space is closed, it is also complete
152442
An annulus in $\mathbb R^2$ is path connected
15865
Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected.
116069
Is every metric space a subspace of some connected metric space?
96781
connected manifolds are path connected
33859
How to show that a disjoint sum of metric spaces is metrizable?