Is a simply connected set connected?

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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Path connected $\Rightarrow$ connected.(math.stackexchange.com/questions/127915/…) –  John Ma Dec 13 '13 at 6:28
> 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 7 '14 at 14:02