Topological space

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Let $X$ be a set and suppose a set $\tau \subset \mathcal{P}(X)$. If $O \in \tau$ then we say that $O$ is an open set of $\tau$ and $X \setminus O$ is a closed set of $\tau$.

  1. $\emptyset, X \in \tau$
  2. If $\mathcal{O} \subset \tau$ then $\bigcup \mathcal{O} \in \tau$
  3. If $\{C_1,C_2, \ldots, C_n \}$ is a finite collection of closed sets, then $\bigcap_{n=1}^n C_n $ is a closed set.

If properties 1-3 hold, then we say that the ordered pair $(X,\tau)$ is a topological space.