Difference between revisions of "Characteristic property of the subspace topology"

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==Proof==
 
==Proof==
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{{Requires proof|grade=A|msg=Already done on task page, just copy and paste! '''SEE [[Task:Characteristic property of the subspace topology|HERE]] PROOF IS ALREADY DONE'''}}
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==See also==
 
==See also==
 
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Latest revision as of 23:25, 25 September 2016

Stub grade: A
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I've already written the proof in a task page so it should be easy to put this all together

Statement

Diagram
Let (X,J) be a topological space and let (S,JS) be any subspace of (X,J)[Note 1]. The characteristic property of the subspace topology[1] is that:
  • Given any topological space (Y,K) and any map f:YS we have:
    • (f:YS is continuous)(iSf:YX is continuous)

Where iS:SX given by iS:ss is the canonical injection of the subspace topology (which is itself continuous)[Note 2]

Proof

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The message provided is:
Already done on task page, just copy and paste! SEE HERE PROOF IS ALREADY DONE

See also


TODO: Put some links here


Notes

  1. Jump up This means SP(X), or SX of course
  2. Jump up This leads to two ways to prove the statement:
    1. If we show iS:SX is continuous, then we can use the composition of continuous maps is continuous to show if f continuous then so is iSf
    2. We can show the property the "long way" and then show iS:SX is continuous as a corollary

References

  1. Jump up Introduction to Topological Manifolds - John M. Lee