Download Continuity Subspace Topology

Continuity
Subspace Topology
Def. Let A ⊂ X where X is a topological space. The subspace
or relative topology on A is defined by: V is open in A if and only
if V = U ∩ A for some open set U in X .
Read pages 48 – 50.
Def. A function f : X → Y , where X and Y are topological
spaces, is continuous at p if whenever U is an open set in Y
containing f (p), there is an open set V in X containing p with
f (V ) ⊂ U
Check that this is a topology.
Note: This is the coarsest topology making the inclusion from A
into X continuous.
Def. A function f : X → Y , where X and Y are topological
spaces is continuous if it is continuous at each point.
Examples:
Theorem: A function f : X → Y , where X and Y are topological
spaces is continuous if and only if whenever U is open in Y ,
f −1 (U ) is open in X .
Examples:
Mth 430 – Winter 2006
Continuity, Subspace Topology
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Mth 430 – Winter 2006
Continuity, Subspace Topology
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