Exercise 2.6

Verify that the discrete topology is a topology.

Ian_mi
$$\emptyset \subseteq X$$ and $$X \subseteq X$$ so $$\emptyset, X \in \mathcal{P}(X)$$.

Let $$I$$ be a collection of sets in $$\mathcal{P}(X)$$. Let $$x \in \bigcup_{U \in I} U$$. Then $$\exists U \in I$$ such that $$x \in U \subseteq X$$. Therefore $$\bigcup_{U \in I} U \subseteq X$$ so $$\bigcup_{U \in I} U \in \mathcal{P}(X)$$.

Let $$A, B \in \mathcal{P}(X)$$. Then $$A \cap B \subseteq A \subseteq X$$ so $$A \cap B \in \mathcal{P}(X)$$. Thus $$(X,\mathcal{P}(X))$$ is a topological space.

Ian mi 00:19, March 15, 2011 (UTC)Ian_mi

'''Cool. --Steven.clontz 19:06, March 17, 2011 (UTC)'''