Does every connected metrizable space have uncountably many distinct open sets, each of which generating a connected subspace?
108052
Does any connected metrizable space with more than one element have uncountably many open sets, each generating a connected subspace?
11383
Disjoint compact sets in a Hausdorff space can be separated
116494
What happens to the connectedness of $\mathbb R^2$ when countable many points are removed?
51710
Is there an infinite connected topological space such that every space obtained by removing one point from it is totally disconnected?
144383
The closure of a connected set in a topological space is connected
39010
Metrization of the cofinite topology
3636
Are any two uncountable sets similar to each other?
66895
A condition for the image of a metrizable space to be metrizable
39746
How to prove that given set is a connected subset of the space of matrices?
18572
Union of a family of connected pairwise not disjoint subspaces is connected
147793
A variation on the induced topology (subspace topology)
119649
Every metrizable space with a countable dense subset has a countable basis
62911
Let $(X,\tau)$ be a connected topological space, $Y\subset X$ connected and $U\subset X\setminus Y$ clopen. Prove that $ Y \cup U$ is connected
48557
Product of connected spaces
134073
Definitions of connected space
109374
Is the Discrete Topology on $X$ the Only One Containing All Infinite Subsets of $X$?
136106
Is every open subset of $ \mathbb{R} $ uncountable?