Download Problem Set 1

Problem Set # 1 / MATH 16200
Due: Thursday, January 21, 2016
1. (a) Suppose A ⊂ X ⊂ C. Prove that the collection of subsets V of X that can be written as V = X ∩U
for some open subset U of C forms a topology on X. In other words, prove that the subspace
topology on X is indeed a topology.
(b) Suppose Y ⊂ X ⊂ C. We can form a topology on Y in two ways. The first way is to first form the
subspace topology on X, as a subspace of C, and then, using that topology, form the subspace
topology on Y , as a subspace of X. The second way is to just ignore X altogether and form the
subspace topology on Y directly as a subspace of C. Prove that these two topologies are in fact
one and the same.
2. Let A ⊂ C and let B 0 ⊂ B ⊂ C.
(a) Suppose f : A → B is continuous and f (A) ⊂ B 0 . Define g : A → B 0 by g (a) := f (a) for each
a ∈ A. Is g continuous?
(b) Suppose f : A → B 0 is continuous. Define g : A → B by g (a) := f (a) for each a ∈ A. Is g
continuous?
3. Let X be any topological space, and let Y ⊂ X. Define f : Y → X by f (y) := y for each y ∈ Y . Prove
that f is continuous.
4. Let X,Y , and Z be topological spaces, and let f : X → Y and g : Y → Z both be continuous. Show
that g ◦ f : X → Z is also continuous. In other words, the composition of two continuous functions is
continuous.
5. Suppose R is a region in C, f : R → C, and f (R) is a finite set. Can f be continuous? If yes, what
are the conditions on f for it to be continuous?
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