- #1
Carl140
- 49
- 0
Homework Statement
Let X be a topological space, a subset S of X is said to be locally closed if
S is the intersection of an open set and a closed set, i.e
S= O intersection C where O is an open set in X and C is a closed set in X
Prove that if M,N are locally closed subsets then M union N is locally closed.
The Attempt at a Solution
So M = O intersection C and N = O' intersection C' where O, O' are open sets in X and
C, C' are closed sets in X.
It follows that M union N = (O intersection C) U (0' intersection C').
From h ere I played with this expression a while using distributive laws but got stuck,
somewhere I end up with the union of a closed set and an open set, i.e O U C, but I think
we cannot said anything about this particular union. Maybe I'm missing some useful
set-theoretical identity. Can you please help?