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Topology : Past Exam Paper

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B5. (a) Define a homomorphism between topological spaces X and Y. Define what is meant by a topological invariant.
(b) State what it means for a map f X -?> Y to be open. Show that a continuous open bijection is a homomorphism.
(c) (i) Recall that Fr E, the frontier of a subset E of a topological space X isdefined as Efl(X?E). Prove that E=EUFrE.
(ii) A topological space is 0 dimensiona1 if and only if whenever x C
V and V is open, there is an open set U with empty frontier such
that x e U C V. Show that the rationals in the relative topology as
a subset of JR with the usual topology is a 0-dimensional set.
(iii) Show that being 0-dimensional is a topological invariant.

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a. A homomorphism between topological space and is a continuous bijection such that is also continuous. A topological invariant of a space is a property that depends only on the topology of the space, i.e. it is shared by any topological space homeomorphic to .
b. Proof:
is open means that for any open ...

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