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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.