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For connected subsets C and D of a topolicical space T where $C^o \cap D^o = \emptyset $ is $C \cup D $ disconnected?

I think it is as C and D are open as they are subsets of a topological space so their inrerior is equal to themselves..

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up vote 6 down vote accepted

You can't tell anything about the connectedness of $C \cup D$ from the information given. Consider $\mathbb{R}$ with the usual topology.

The sets $[-1,0],[0,1]$ are connected, and $[-1,0]^\circ \cap [0,1]^\circ = (-1,0) \cap (0,1) = \emptyset$, however $[-1,0] \cup [0,1] = [-1,1]$ is connected.

On the other hand, $(-1,0), (0,1)$ are disjoint connected open sets in $\mathbb{R}$, and thus their union is not connected.

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Consider the real line. Let $C$ be the rationals and $D$ be the irrationals. Their interiors are both void, but $C\cup D$ is the entire (connected) real line.

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