proof of characterization of connected compact metric spaces.

First we prove the right-hand arrow: if X is connected then the property stated in the TheoremMathworldPlanetmath holds. This implicationMathworldPlanetmath is true in every metric space X, without additional conditions.

Let us denote with Aε the set of all points zX which can be joined to x with a sequence of points p1,,pn with p1=x, pn=z and d(pi,pi+1)<ε. If zAε then also Bε(z)Aε since given wBε(z) we can simply add the point pn+1=w to the sequence p1,,pn. This immediately shows that Aε is an open subset of X. On the other hand we can show that Aε is also closed. In fact suppose that xnAε and xnx¯X. Then there exists k such that x¯Bε(xk) and hence x¯Aε by the property stated above. Since both Aε and its complementary set are open then, being X connected, we conclude that Aε is either empty or its complementary set is empty. Clearly xAε so we conclude that Aε=X. Since this is true for all ε>0 the first implication is proven.

Let us prove the reverse implication. Suppose by contradictionMathworldPlanetmathPlanetmath that X is not connected. This means that two non-empty open sets A,B exist such that AB=X and AB=. Since A is the complementary set of B and vice-versa, we know that A and B are closed too. Being X compact we conclude that both A and B are compact sets. We now claim that


Suppose by contradiction that δ=0. In this case by definition of infimumMathworldPlanetmath, there exist two sequences akA and bkB such that d(ak,bk)0. Since A and B are compact, up to a subsequence we may and shall suppose that akaA and bkbB. By the continuity of the distance function we conclude that d(a,b)=0 i.e. a=b which is in contradiction with the condition AB=. So the claim is proven.

As a consequence, given ε<δ it is not possible to join a point of A with a point of B. In fact in the sequence p1,,pn there should exists two consecutive points pi and pi+1 with piA and pi+1B. By the previous observation we would conclude that d(pi,pi+1)δ>ε.

Title proof of characterizationMathworldPlanetmath of connected compact metric spaces.
Canonical name ProofOfCharacterizationOfConnectedCompactMetricSpaces
Date of creation 2013-03-22 14:17:06
Last modified on 2013-03-22 14:17:06
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 4
Author paolini (1187)
Entry type Proof
Classification msc 54A05