# Monthly Archives: February 2013

## Topology: Finite Intersection Property (Omake)

The whole point of this article is the following seemingly trivial observation. Theorem. A topological space X is compact if and only if it satisfies the finite intersection property (F.I.P.): if is a collection of closed subsets of X such that … Continue reading

## Topology: More on Compact Spaces

In the previous article, we defined compact spaces as those where every open cover has a finite subcover, i.e. if then we can find a finite set of indices such that On an intuitive level, one should imagine a compact … Continue reading

## Topology: Sequentially Compact Spaces and Compact Spaces

We’ve arrived at possibly the most confusing notion in topology/analysis. First, we wish to fulfil an earlier promise: to prove that if C is a closed and bounded subset of R and f : R → R is continuous, then f(C) is closed and bounded. [ As … Continue reading

Posted in Notes | | 4 Comments

## Topology: Complete Metric Spaces

[ This article was updated on 8 Mar 13; the universal property is now in terms of Cauchy-continuous maps. ]  On an intuitive level, a complete metric space is one where there are “no gaps”. Formally, we have: Definition. A … Continue reading

Posted in Notes | | 11 Comments

## Topology: Hausdorff Spaces and Dense Subsets

Hausdorff Spaces Recall that we’d like a condition on a topological space X such that if a sequence converges, its limit is unique. A sufficient condition is given by the following: Definition. A topological space X is said to be Hausdorff if … Continue reading

Posted in Notes | | 4 Comments

## Topology: Product Spaces (II)

The Box Topology Following an earlier article on products of two topological spaces, we’ll now talk about a product of possibly infinitely many topological spaces. Suppose is a collection of topological spaces indexed by I, and we wish to define … Continue reading

## Topology: Interior

Let Y be a subset of a topological space X. In the previous article, we defined the closure of Y as the smallest closed subset of X containing Y. Dually, we shall now define the interior of Y to be the largest open subset contained in … Continue reading

Posted in Notes | | 2 Comments

## Topology: Closure

Suppose Y is a subset of a topological space X. We define cl(Y) to be the “smallest” closed subset containing Y. Its formal definition is as follows. Let Σ be the collection of all closed subsets containing Y. Note that , so Σ is not empty. … Continue reading