Characteristic property of the subspace topology
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Contents
[hide]Statement
- Given any topological space (Y,K) and any map f:Y→S we have:
- (f:Y→S is continuous)⟺(iS∘f:Y→X is continuous)
Where iS:S→X given by iS:s↦s is the canonical injection of the subspace topology (which is itself continuous)[Note 2]
Proof
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See also
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Notes
- Jump up ↑ This means S∈P(X), or S⊆X of course
- Jump up ↑ This leads to two ways to prove the statement:
- If we show iS:S→X is continuous, then we can use the composition of continuous maps is continuous to show if f continuous then so is iS∘f
- We can show the property the "long way" and then show iS:S→X is continuous as a corollary
References
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