Math 434/542
Fall 2015
Test 1, 9-28-15
Carter
1. Let X denote a set. Let P(X) = {A ⊂ X}. Suppose that T ⊂ P(X). Define the term
T is a topology on X.
2. In case X is finite, say X has n elements. Indicate why P(X) has 2n elements.
3. Suppose that X is infinite. Let T = {A ⊂ X| X \ A is finite }. Here X \ A = {x ∈
X| x ∈
/ A}. Show that T is a topology on X. (If you have trouble with this because
you could not solve #1, please ask for the definition.)
f
4. Suppose (X, T ) and (Y, V) are topological spaces and Y ← X denotes a function.
Define f is continuous on X.
5. Describe the open sets in the usual topology on the real line R. This is the topology
induced by the metric d(x, y) = |y − x| on R.
6. Consider the identity function on R. Also consider the usual and finite complement
topology (problem #3). The identity function is continuous in one direction. Which
one?
7. Let X denote a set; and A 7→ A a function from P(X) to P(X) such that
(a) ∅ = ∅;
(b) A ⊂ A;
(c) A = A;
(d) A ∪ B = A ∪ B.
Show that
T = {X \ A| A ∈ P(X)}
is a topology on X.
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