Download IN-CLASS PROBLEM SET (1) Find a continuous surjection f : R → {a

IN-CLASS PROBLEM SET
JUNE 13, 2013
(1) Find a continuous surjection f : R → {a, b} for each of the following topologies
on {a, b}, or explain why no such function exists. In all cases assume R has the
standard topology.
(a) the discrete topology
(b) {∅, {a}, {a, b}}
(c) the indiscrete topology
(2) Define a relation ∼
= on the set of topological spaces as follows: for X, Y topological
spaces, we say X ∼
= Y if there is a homeomorphism f : X → Y . Prove ∼
= is an
equivalence relation.
(3) Prove that the − δ definition of continuity for functions f : R → R is equivalent
to the open set definition (assuming here the standard topology on R).
(4) Suppose X, Y, Z are topological spaces and suppose f : X → Y g : Y → Z are
continuous functions. Prove g ◦ f is a continuous functon.
(5) Suppose f, g : X → R are continuous functions, where X is some topological space
and R has the standard topology. Prove f + g is a continuous function.
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