Author Archives: limsup

Topology: Separation Axioms

Posted on March 15, 2013 by limsup

Motivation The separation axioms attempt to answer the following. Question. Given a topological space X, how far is it from being metrisable? We had a hint earlier: all metric spaces are Hausdorff, i.e. distinct points can be separated by two … Continue reading →

Posted in Notes | Tagged advanced, Hausdorff, metrisable topology, normal, regular, separation axioms, T1, T2, T3, T4, topology, urysohn's lemma, urysohn's metrisation theorem | Leave a comment

Topology: Locally Connected and Locally Path-Connected Spaces

Posted on March 10, 2013 by limsup

Locally Connected Spaces Recall that each topological space X is the set-theoretic disjoint union of its connected components, but in general (e.g. for X=Q) fails to be the topological disjoint union. The problem is that the connected components in general aren’t open … Continue reading →

Posted in Notes | Tagged advanced, connected components, connected spaces, locally connected, locally path-connected, path-connected components, path-connected spaces, topology, totally disconnected spaces | 4 Comments

Topology: Path-Connected Spaces

Posted on March 7, 2013 by limsup

A related notion of connectedness is this: Definition. A path on a topological space X is a continuous map The path is said to connect x and y in X if f(0)=x and f(1)=y. X is said to be path-connected if any two points … Continue reading →

Posted in Notes | Tagged advanced, connected components, connected spaces, path-connected components, path-connected spaces, topology | 2 Comments

Topology: Connected Spaces

Posted on March 5, 2013 by limsup

Let X be a topological space. Recall that if U is a clopen (i.e. open and closed) subset of X, then X is the topological disjoint union of U and X–U. Hence, if we assume X cannot be decomposed any further, there’re no non-trivial clopen subsets of X. … Continue reading →

Posted in Notes | Tagged advanced, box topology, connected components, connected spaces, intermediate value theorem, product topology, topology, totally disconnected spaces | Leave a comment

Topology: One-Point Compactification and Locally Compact Spaces

Posted on March 3, 2013 by limsup

Compactifications There’re lots of similarities between completeness and compactness, beyond the superficial resemblance of the words. For example, a closed subset of a compact (resp. complete) space is also compact (resp. complete). Two differences though: compactness is a topological concept … Continue reading →

Posted in Notes | Tagged advanced, Alexandroff extensions, compact spaces, compactifications, Hausdorff, locally compact spaces, one-point compactification, topology | 6 Comments

Topology: Finite Intersection Property (Omake)

Posted on February 28, 2013 by limsup

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 →

Posted in Notes | Tagged advanced, compact spaces, finite intersection property, four colour theorem, topology | Leave a comment

Topology: More on Compact Spaces

Posted on February 26, 2013 by limsup

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 →

Posted in Notes | Tagged advanced, alexander's subbase theorem, axiom of choice, compact spaces, complete metric spaces, subbases, topology, tychonoff's theorem, uniform continuity, zorn's lemma | Leave a comment

Topology: Sequentially Compact Spaces and Compact Spaces

Posted on February 24, 2013 by limsup

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 | Tagged compact spaces, metric spaces, nets, sequences, sequentially compact spaces, subnets | 4 Comments

Topology: Complete Metric Spaces

Posted on February 21, 2013 by limsup

[ 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 | Tagged advanced, cauchy sequences, complete metric spaces, completion, metric spaces, topology, universal properties | 11 Comments

Topology: Hausdorff Spaces and Dense Subsets

Posted on February 18, 2013 by limsup

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 | Tagged advanced, continuity, dense subsets, Hausdorff, metric spaces, nets, topology | 4 Comments