Difference between revisions of "Open and closed maps"
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(Created page with "Due to the parallel definitions and other similarities there is little point to having separate pages. ==Definition== ===Open map=== A <math>f:(X,\mathcal{J})\rightarrow (Y,\...") |
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Revision as of 07:33, 8 April 2015
Due to the parallel definitions and other similarities there is little point to having separate pages.
Contents
[hide]Definition
Open map
A f:(X,J)→(Y,K) (which need not be continuous) is said to be an open map if:
- The image of an open set is open (that is ∀U∈J[f(U)∈K])
Closed map
A f:(X,J)→(Y,K) (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
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Theorem: Given a map f:(X,J)→(Y,K) that is continuous, surjective and an open or closed map, then the topology K on Y is the same as the Quotient topology induced on Y by f
