## Transcribed Text

1. Recall that a subset A of a topological space (X, T) is closed provide X − A ∈ T. In
(X, T), prove that
a) ∅, X are closed
b) Finite unions of closed sets are closed
c) Arbitrary intersections of closed sets are closed.
2. Suppose that Y is a subspace of the topological space X. We say A ⊂ Y is closed in
the subspace topology Y provided Y − A is open in Y .
a) Show by example that a closed subset of the subspace Y is not necessarily closed a
subset of X.
b) Prove that A is closed in the subspace topology on Y if and only if A = Y ∩ C for
some closed subset C ⊂ X.
3. Let X be a topological space and let A ⊂ X. Then
- The interior of A, denoted Å, is the union of all open subsets contained in A.
- The closure of A, denoted A¯, is the intersection of all closed sets containing A.
- The boundary of A, denoted ∂A, is the intersection A¯ ∩ X − A.
Observe that
Å ⊂ A ⊂ A. ¯
4. Consider A = [0, 1] ⊂ R. What are the interior, closure and boundary of A ? Repeat
the question for the subset B = {1, 1/2, 1/4, 1/8, . . .}
5. Prove the following statements:
a) Å is open.
b) A¯ is closed.
c) Å ∩ ∂A = ∅.
d) A¯ = Å ∪ ∂A.
6. Let A be the subset of the topological space X. Prove that x ∈ A¯ if and only if,
for every open set U containing x, U ∩ A 6= ∅. (Hint: the contrapositive in both directions.)
7. Let A be a subset of the topological space X, and suppose β is a basis for the topology
on X. Prove that x ∈ A¯ if and only if, for every B ∈ β containing x, B ∪ A 6= ∅ (Hint:
use the previous result.)
8. Let A be a subset of a topological space X. Then x ∈ X is a limit point for A
if, for every open set U containing x, U ∩ A contains at least one element y not equal to
x itself.
a) What are the limit points of (0, 1] ⊂ R ?
b) What are the limit points of {
n
n+1 | n ∈ N} ⊂ R ?
c) What are the limit points of S
1 ⊂ R
2
? Here S
1
is the unit circle in R
2
; assume R
2 has
the usual topology (generated by the δ-ball basis).
d) What are the limit points of Z ⊂ R ?
e) What are the limit points of {x ∈ R|x > 0} ?
19. Given the subset A of the topological space X, let A0 denote the set of limit points of
A. Prove that
A¯ = A
0 ∪ A.
As a corollary, prove that A ⊂ X is closed if and only if A0 ⊂ A (i.e. A contains all of its
limit points).
10. Show that the topological space X is not connected if it contains a proper subset
A which is both open and closed. Then use this result to characterize connected topological spaces.
11. We dened a path-connected topological space X to be any space in which, for
any pair of points x, y, there exists a continuous function f : [0, 1] → X with f(0) = x
and f(1) = y.
Prove that if X is path-connected and g : X → Y is a continuous function between topological spaces, then the image f(X) of X is path connected.
12. Consider the topological space R
2
in the usual topology (generated by the δ-ball
basis). Recall that a path connected space is connected (but the converse is false).
a) Let L ⊂ R
2 be any straight line. Is R
2 − L connected ?
b) Is R
2 − S
1
connected ?
c) Let A be any nite subset of R
2
. Is R
2 − A connected ?
d) Z × Z is the subset of R
2
consisting of all points with integer coordinates (aka the
integer lattice). Is R
2 − Z × Z connected ?

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