# Proving Locally Closed Property of M Union N in Topological Spaces

• Carl140
In summary, the problem statement defines locally closed subsets in a topological space as the intersection of an open set and a closed set. The task is to prove that the union of two locally closed subsets is also locally closed. A potential counterexample in \mathbb{R} is given, but further clarification may be needed.
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

Carl140 said:
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.

Last edited:

## 1. What does it mean for a set to have the locally closed property?

The locally closed property means that a set can be written as the intersection of an open set and a closed set. In other words, it is both open and closed in a given topological space.

## 2. How do you prove that a set has the locally closed property?

To prove that a set has the locally closed property, you must show that it can be written as the intersection of an open set and a closed set. This can be done by showing that the set is both open and closed in the given topological space.

## 3. What is the significance of proving the locally closed property of a set?

Proving the locally closed property of a set is important in topology because it allows us to understand the structure of a topological space. It also helps us to identify sets that are both open and closed, which can have important implications in various mathematical proofs and applications.

## 4. Can the locally closed property of a set hold in all topological spaces?

No, the locally closed property may not hold in all topological spaces. It depends on the specific structure and properties of the topological space. For example, in a discrete topological space, all subsets are both open and closed, so the locally closed property holds for all sets.

## 5. How does the locally closed property relate to other topological properties?

The locally closed property is related to other topological properties such as being connected and being compact. In fact, a topological space is connected if and only if it has no nonempty subsets with the locally closed property. Similarly, a topological space is compact if and only if every subset with the locally closed property is also compact.

• Calculus and Beyond Homework Help
Replies
2
Views
859
• Calculus and Beyond Homework Help
Replies
12
Views
2K
• Calculus and Beyond Homework Help
Replies
12
Views
2K
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
20
Views
3K
• Calculus and Beyond Homework Help
Replies
12
Views
2K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
20
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
761