proof of characterization of connected compact metric spaces.
Let us denote with the set of all points which can be joined to with a sequence of points with , and . If then also since given we can simply add the point to the sequence . This immediately shows that is an open subset of . On the other hand we can show that is also closed. In fact suppose that and . Then there exists such that and hence by the property stated above. Since both and its complementary set are open then, being connected, we conclude that is either empty or its complementary set is empty. Clearly so we conclude that . Since this is true for all the first implication is proven.
Let us prove the reverse implication. Suppose by contradiction that is not connected. This means that two non-empty open sets exist such that and . Since is the complementary set of and vice-versa, we know that and are closed too. Being compact we conclude that both and are compact sets. We now claim that
Suppose by contradiction that . In this case by definition of infimum, there exist two sequences and such that . Since and are compact, up to a subsequence we may and shall suppose that and . By the continuity of the distance function we conclude that i.e. which is in contradiction with the condition . So the claim is proven.
As a consequence, given it is not possible to join a point of with a point of . In fact in the sequence there should exists two consecutive points and with and . By the previous observation we would conclude that .
|Title||proof of characterization of connected compact metric spaces.|
|Date of creation||2013-03-22 14:17:06|
|Last modified on||2013-03-22 14:17:06|
|Last modified by||paolini (1187)|