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?
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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.