## Transcribed Text

I. In class, we considered a pair of functions: /: R ➔ R, /(:r:) = sin(:r:); and g: R ➔ R2
,
g(x) = (cos(x),sin(x)).
(a) Compute 1- 1
( ½)-
(b) Compute g- 1
((1,0)).
2. How many distinct topologies are there on the set X = { a, b, c}? For the purposes of this
problem, consider two topologies T and T' distinct if, as collections of subsets of X, T 'I- T' •
For example, T = {0, {a}, X} and T' = {a, {b}, X} are c.li1;;ti11ct topologies.
3. Let T and T' be topologies on X. Prove or give a counterexample:
( a) T n T' is also a topology on X.
(b) TU T' is also a topology on X.
•
5. Let
/3 = { la, b) I a, b E R, a < b}.
{a) Prove that /3 defines a basis for a topology on IR.
(b) If T is the u.suaJ topology on JR and T' is the topology on JR induced by /3, how do T and
T' compare·! Explain.
6. Consider the function f : R --; JR defined by
f(x) = {X X < 1
x+l xl
Show that f is not continuous if JR is given the usual topology, but becomes continuous using
the topology T' from Problem f>.
7. Suppose (X, T) is a topological space and.that /3 is a basis for T. Prove that if Y c X has the
subspace topology Ty inherited from X, then
{Jy = { B n Y I B E f3}
is a basis for Ty.
8. Let R have the usual topology.
(a) Prove that the subspace topology on Z inherited from R coincides with the discrete
the topology on Z.
9. Let f : X ➔ Y be a function between topological spaces X a.nd Y. Prove that / is continuous
if and only if 1-1 ( C) C X is closed whenever C C Y is closed.

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