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## Homework Statement

Let ##X## be a limit point compact space. If ##X## is a subspace of the Housdorff space ##Z##, does it follow that ##X## is closed in Z?

## Homework Equations

A space ##X## is said to be limit point compact if every infinite subset of ##X## has a limit point.

## The Attempt at a Solution

If ##X## is finite it is closed so suppose it is infinite then it has a limit point in ##X## however this does not exclude the posibility of ##X## having a limit point in ##Z## outside ##X## which implies that it is not closed in ##Z##.

Is this correct? In which case a counter example is needed.