Vipul: Created page with '{{topospace property non-implication| stronger = connected space| weaker = locally connected space}} ==Statement== It is possible for a topological space to be a [[connecte…'

2009-12-25T07:09:17Z

Created page with '{{topospace property non-implication| stronger = connected space| weaker = locally connected space}} ==Statement== It is possible for a topological space to be a [[connecte…'

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{{topospace property non-implication|
stronger = connected space|
weaker = locally connected space}}

==Statement==

It is possible for a [[topological space]] to be a [[connected space]] but ''not'' a [[locally connected space]].

==Definitions used==

===Connected space===

{{further|[[connected space]]}}

A [[topological space]] is termed connected if it cannot be expressed as a disjoint union of two nonempty open subsets.

===Locally connected space===

{{further|[[Locally connected space]]}}

A [[topological space]] <math>X</math> is termed locally connected if, for every point <math>x \in X</math> and every open subset <math>U</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>V</math> of <math>X</math> such that <math>x \in V</math>, <math>\overline{V}\subseteq U</math>, and <math>V</math> is a [[connected space]].

==Proof==

The general idea behind counterexamples is that the ''connecting apparatus'' between a point and points very ''close'' to it is via points that are very far from it. Most of these counterexamples are also counterexamples for the related fact that [[path-connected not implies locally path-connected]], where points ''close by'' can be connected only via paths that go through points that are ''far away''.

Here are some counterexample spaces (more elaboration needed):

* The [[particular example::infinite broom]] and [[particular example::closed infinite broom]]. Both of these are connected spaces, and the latter is also a [[path-connected space]]. However, neither of these is locally connected.
* The [[particular example::topologist's sine curve]] is a [[connected space]] but not a [[locally connected space]].
* The [[particular example::comb space]] is a connected space -- in fact, it is a [[contractible space]]. However, it is ''not'' locally connected.

Connected not implies locally connected - Revision history

Vipul: Created page with '{{topospace property non-implication| stronger = connected space| weaker = locally connected space}} ==Statement== It is possible for a topological space to be a [[connecte…'