connected sets dont have subsets that are both open and closed
44673
$M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time
80524
Path-connected and locally connected space that is not locally path-connected
41611
Open and closed set in n by n matrices space.
70048
In which topologies do open sets maintain open under countable or arbitrary intersection?
8015
Compact set plus a closed set is a closed set
148992
A finite set is closed
79424
Don't understand proof that interior of a set is open
88301
What are all of the connected subsets of $\mathbb{Q}$?
137460
How should I think of an open vs. closed set?
39746
How to prove that given set is a connected subset of the space of matrices?
146016
no cycles if and only if has $n-m$ connected components
46698
Showing a set is closed under inverses
127900
Interior of closure of open set in topological space
99678
Is a sphere a closed set?
143457
Is product of two closed sets closed?
27998
Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.
108052
Does any connected metrizable space with more than one element have uncountably many open sets, each generating a connected subspace?