Show a metrix space is connected iff for every non-empty proper subset the boundary is non-empty
58285
Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$
108052
Does any connected metrizable space with more than one element have uncountably many open sets, each generating a connected subspace?
18572
Union of a family of connected pairwise not disjoint subspaces is connected
59424
Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
26919
Intersection of Connected Sets
3649
Closed unit interval is connected proof
106298
Prove that some topology is not metrizable
135025
Axiomatization of a Topology using the Boundary
158850
Is it true that a boundary of a simply connected and bounded set is connected in $\mathbb{C}$?
116538
Proof of Connected Diagrams
55486
Topological proof to show a complement set of a set is polygonally connected
42811
Product of path connected spaces is path connected
66895
A condition for the image of a metrizable space to be metrizable
42494
The boundary is a closed set
48557
Product of connected spaces
157218
If $E\subseteq X$ is connected, $E\cap A\neq\emptyset$ and $E\cap(X\setminus A)\neq\emptyset$ then $E\cap\partial A\neq\emptyset$
32070
Prove that the Interior of the Boundary is Empty