A description/proof of that if union of disjoint subsets is closed, each subset is not necessarily closed
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
Target Context
- The reader will have a description and a proof of the proposition that if the union of some disjoint subsets is closed, each subset is not necessarily closed.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any topological space, \(T\), and some disjoint subsets, \(\{S_i| S_i \subseteq T\}\), such that the union, \(S := \cup_i S_i\), is closed, each \(S_i\) is not necessarily closed.
2: Proof
A counterexample will suffice. Think of the \(\mathbb{R}\) Euclidean topological space, and 2 subsets, \([-1, 0], (0, 1]\), which are disjoint and the union is \([-1, 1]\), closed.
