Proving Locally Closed Property of M Union N in Topological Spaces

[Carl140]

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?

 

[Unco]

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.

Are you certain you have the problem statement written correctly, Carl? Consider $$\mathbb{R}$$ with open sets $$\emptyset$$, $$\mathbb{R}$$ and $$(-\infty,a)$$ for $$a\in\mathbb{R}$$ for a counterexample.