- #1

Bashyboy

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## Homework Statement

If ##C## is a connected space in some topological space ##X##, then the closure ##\overline{C}## is connected.

## Homework Equations

## The Attempt at a Solution

Suppose that ##\overline{C} = A \cup B## is separation; hence, ##A## and ##B## are disjoint and do not share limit points, which means ##A \cap \overline{B}## and ##\overline{A} \cap B## are both empty. Since ##C## is connected and is contained in ##\overline{C} = A \cup B##, it must be contained in exactly one of the two partitions. WLOG, suppose that ##C \subseteq A##. Then ##C \subseteq A \subseteq \overline{C}## implies ##\overline{A} = \overline{C}##, and therefore ##\emptyset = \overline{A} \cap B = \overline{C} \cap B## implies ##B= \emptyset. Hence, ##\overline{C}## is connected.

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