Given two open sets, show their sum is open
123897
Sum of two open sets is open?
70048
In which topologies do open sets maintain open under countable or arbitrary intersection?
130023
Prove that the sum of two continuous functions is continuous
51260
Sum of sets of measure zero
157753
Show that any set in a metric space can be written as the intersection of open sets
152193
The intersection of a closure with an open set
146664
The union of two linearly independent subsets
33014
Bijection between an open and a closed interval
106434
Prove that for a dense set $A$ and an open set $U$, $U\subset \overline{A\cap U}$
132541
Showing ${1\over n}\sum|S_i|=O(\sqrt n)$ for $S_i\subset [n]$, $|S_i\cap S_i|\le 1$ for $i\ne j$
46698
Showing a set is closed under inverses
88982
Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
42145
Prove that the only sets in $R$ which are both open and closed are the empty set and $R$ itself.
58004
showing a collection of sets contain all closed sets
123149
Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$
142779
How to show distance between boundary of an open set and a compact subset is positive?
39020
Why is empty set an open set?