(a) Let X and Y be topological spaces, and let X xY be the corresponding product space. Define the projections px : X × Y → X and py : X × Y → Y by px(x, y) = x and py(x, y) = y. Prove that px and py are continuous. (b) Show that the product topology is the coarsest topology for which both functions px and py are continuous. That is, show that if T is the product topology on X × Y and T' is a topology on X x Y such that px and py are continuous, then T CT'.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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(a) Let X and Y be topological spaces, and let X × Y be the corresponding product space.
Define the projections
px : X x Y → X and py : X x Y →Y
by px (x, y) = x and py (x, y) = y. Prove that px and PY are continuous.
(b) Show that the product topology is the coarsest topology for which both functions px
and py are continuous. That is, show that if T is the product topology on X x Y and
T' is a topology on X × Y such that px and py are continuous, then TCT'.
Transcribed Image Text:(a) Let X and Y be topological spaces, and let X × Y be the corresponding product space. Define the projections px : X x Y → X and py : X x Y →Y by px (x, y) = x and py (x, y) = y. Prove that px and PY are continuous. (b) Show that the product topology is the coarsest topology for which both functions px and py are continuous. That is, show that if T is the product topology on X x Y and T' is a topology on X × Y such that px and py are continuous, then TCT'.
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