User talk:Jrottman

I replied to your question on the topology Wikibook. Please see my answer there.

Topology Expert (talk) 13:27, 9 June 2008 (UTC)

2. How are the concepts of base and open cover related? It seems that every base is an open cover, but not every open cover is a base. But, why are both concepts needed?

The terms base and open cover are not evidently related. Every base is an open cover which is probably the main relation. Take a second countable topological space for instance (secound countable means that the space has a countable base for its topology). Such a space satisfies the property that every open cover has a countable subcover. To prove this we use the countability of the base. Basically, for any open cover, we choose for each element of the space, an element of the open cover containing it and hence a basis element contained in that element of open cover. Therefore, for any open cover, we can generate a open cover of basis elements that is an 'open refinement' (see Wikipedia for definition). From here we can get properties of open covers from properties of the base. If the base is countable, we can generate a countable open cover from the original cover. Does this answer you question?

Topology Expert (talk) 13:28, 9 June 2008 (UTC)