Exercise 2.4

Let $$X$$ be any metric space with metric $$d$$. Show that $$X$$ is a topological space, where the topology $$T$$ consists of $$\emptyset$$ and all subsets of $$X$$ that are open in the sense of Definition 1.7.

Ian_mi
Let $$(X,d)$$ be a metric space. Let $$\mathcal{T}$$ be the collection of open sets in $$X$$. Then $$\emptyset$$ and $$X$$ are open and $$\mathcal{T}$$ is closed under arbitrary unions and finite intersection by Propositions 1.11 and 1.12. Therefore $$(X,\mathcal{T})$$ is a topological space.

Ian mi 22:26, March 14, 2011 (UTC)Ian_mi

'''Looks good. I went ahead and changed $$U$$ to $$\mathcal{T}$$ since it's a topology (and often we'll use $$U$$ as an arbitrary open set, rather than a collection of open sets). --Steven.clontz 04:41, March 15, 2011 (UTC)'''