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A set $X$ is considered connected if there is no separation of the set $X$ into disjoint sets $A,B$ such that $X = A \cup B$, where neither sets ($A$,$B$) contain limit points of each other. Now a simply connected set is path connected and has a trivial fundamental group. My question is does if a space $X$ is simply connected, does it imply connected?

kolonel
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    Path connected $\Rightarrow$ connected.(http://math.stackexchange.com/questions/127915/path-connected-implies-connected-without-knowing-that-0-1-is-connected) –  Dec 13 '13 at 06:28

  • the union of two open disjoint discs are simply connected< It is a disjoint union of simply connected sets, but since it is not path-connected, it cannot be simply connected by definition.

     – eltonjohn Mar 07 '14 at 14:02

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A simply connected set is by definition path-connected (= any two points in it can be connected by a path contained in the set). And a path-connected set is connected (means: not a union of two open sets that have no points in common), so a simply connected set is connected.

Semsem
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eltonjohn
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  • I don't like to start a meta-discussion here, but why did someone give me a down vote? The first statement is a reminder of the definition, and the second statement is an easily exercise. – eltonjohn Mar 07 '14 at 14:26