Math and science::Topology

Closed subsets

Closed subsets, definition

Let $$X = (X, \mathcal{T})$$ be a topological space. A subset $$V \subseteq X$$ is closed iff [...].

The below properties of closed sets can be derived.

Lemma. [hidden as it gives away the below cloze.]

Let $$X = (X, \mathcal{T})$$ be a topological space. Then

1. Whenever $$V_1$$ and $$V_2$$ are closed subsets of $$X$$, then [...] is also closed in X.
2. Whenever $$(V_i)_{i\in I}$$ is a family (finite or not) of closed subsets of $$X$$, then [...] is closed in $$X$$.
3. [something] and [something] are closed subsets of $$X$$.

This lemma follows from the application of de Morgan's laws to the definition of a topological space.