Open and closed maps
Due to the parallel definitions and other similarities there is little point to having separate pages.
Contents
Definition
Open map
A [math]f:(X,\mathcal{J})\rightarrow (Y,\mathcal{K})[/math] (which need not be continuous) is said to be an open map if:
- The image of an open set is open (that is [math]\forall U\in\mathcal{J}[f(U)\in\mathcal{K}][/math])
Closed map
A [math]f:(X,\mathcal{J})\rightarrow (Y,\mathcal{K})[/math] (which need not be continuous) is said to be a closed map if:
- The image of a closed set is closed
TODO: References - it'd look better
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
Theorem: Given a map [math]f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})[/math] that is continuous, surjective and an open or closed map, then the topology [ilmath]\mathcal{K} [/ilmath] on [ilmath]Y[/ilmath] is the same as the Quotient topology induced on [ilmath]Y[/ilmath] by [ilmath]f[/ilmath]
Note: the intuition behind this theorem is fairly obvious, if [ilmath]f[/ilmath] is an open map then it takes open sets to open sets, but the quotient topology is precisely those sets in [ilmath]Y[/ilmath] whose preimage is open in [ilmath]X[/ilmath]
TODO: Come back after writing about how the quotient topology is the strongest topology - rather than doing that proof here
