Difference between revisions of "Continuous map/Claim: continuous iff continuous at every point"
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(Created page with "<noinclude> ==Statement== A map, {{M|f:X\rightarrow Y}} between two topological spaces {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} is ''continuou...") |
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Latest revision as of 16:10, 23 March 2016
Contents
[hide]Statement
A map, f:X→Y between two topological spaces (X,J) and (Y,K) is continuous if and only if it is continuous at every point. Symbolically:
- (∀O∈K[f−1(O)∈J])⟺(∀x0∈X∀N neighbourhood to f(x0)[f−1(N) is a neighbourhood of x0])
Proof
- A quick proof I did on some scrap - click for full version
This requires one or more proofs to be written up neatly and is on a to-do list for having them written up. This does not mean the results cannot be trusted, it means the proof has been completed, just not written up here yet. It may be in a notebook, some notes about reproducing it may be left in its place, perhaps a picture of it, so forth. The message provided is:
See image
Notes
References
