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Consider a topological space $X$ and a subset $Z \subset X$. Assume we are given a continuous injective map (in $X$ topology) $f:Z \to Z$ such that $g: Z \setminus C \to Z$(where $C \subset Z$) and $g\circ f = {\rm id}_Z$ is also continuous. Will $f$ and $g$ remain continuous in the induced topology, i.e. the topology induced on $Z$ by the topology of $X$?

Thank you

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Since the domains of $f$ and $g$ are in $Z$, all the open preimages under them are in $Z$. Thus they remain open in the induced topology, since they don't change by intersection with $Z$. Thus $f$ and $g$ remain continuous.

This seems a bit too easy – am I misunderstanding the question?

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    @Ali: One more comment: The formulation in the question is a bit confusing -- grammatically, "is also continuous" appears to refer to $g\circ f = {\rm id}_Z$, but since ${\rm id}_Z$ is trivially continuous, and since the next sentence speaks of $g$ "remaining" continuous, I assumed that it referred to $g$ being continuous.2011-10-23