Open and closed maps

Due to the parallel definitions and other similarities there is little point to having separate pages.

Definition

Open map

A $$f:(X,\mathcal{J})\rightarrow (Y,\mathcal{K})$$ (which need not be continuous) is said to be an open map if:

• The image of an open set is open (that is $$\forall U\in\mathcal{J}[f(U)\in\mathcal{K}]$$ )

Closed map

A $$f:(X,\mathcal{J})\rightarrow (Y,\mathcal{K})$$ (which need not be continuous) is said to be a closed map if:

• The image of a closed set is closed

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TODO: References - it'd look better

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Importance with respect to the quotient topology

The primary use of recognising an open/closed map comes from recognising a Quotient topology, as the following theorem shows

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Theorem: Given a map $$f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})$$ that is continuous, surjective and an open or closed map, then the topology $\mathcal{K}$ on $Y$ is the same as the Quotient topology induced on $Y$ by $f$

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Note: the intuition behind this theorem is fairly obvious, if $f$ is an open map then it takes open sets to open sets, but the quotient topology is precisely those sets in $Y$ whose preimage is open in $X$

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TODO: Come back after writing about how the quotient topology is the strongest topology - rather than doing that proof here

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See also

• Continuous map
• Topology