Page 69 - DMTH503_TOPOLOGY

Unit 6: Closed Sets and Limit Point

Theorem 1: Let X be a topological space. Then the following conditions hold: Notes
(a)  and X are closed.
(b) Arbitrary intersections of closed sets are closed.
(c) Finite unions of closed sets are closed.
Proof:

(a)  and X are closed because they are the complements of the open sets X and , respectively.
A
(b) Given a collection of closed sets   , we apply De Morgan’s law,
  J
X  A  X A .
 
 J  J
Since the sets X A  are open by definition, the right side of this equation represents an
arbitrary union of open sets, and is thus open. Therefore, A is closed.

(c) Similarly, if A is closed for i = 1, …, n, consider the equation
i
n n
X  A   X A . 
i i


i 1 i 1
The set on the right side of this equation is a finite intersection of open sets and is therefore
open. Hence  A is closed.
i
Theorem 2: Let Y be a subspace of X. Then a set A is closed in Y if and only if it equals the
intersection of a closed set of X with Y.
Proof: Assume that A = C  Y, where C is closed in X. Then X  C is open in X, so that
(X  C)  Y is open in Y, by definition of the subspace topology. But (X  C)  Y = Y  A. Hence
Y  A is open in Y, so that A is closed in Y. Conversely, assume that A is closed in Y. Then Y  A
is open in Y, so that by definition it equals the intersection of an open set U of X with Y. The set
X  U is closed in X and A = Y  (X  U), so that A equals the intersection of a closed set of X with
Y, as desired.

Example 3: Let (Y, U)  (X, T) and A  Y.
Then A is U-closed iff A = F  Y for some T closed set F.
or

A is U-closed iff A is the intersection of Y and a T-closed F.
Solution: Let (Y, U)  (X, T) and A  Y, i.e. (Y, U) is subspace of (X, T).
To prove that A is U-closed iff
A = F  Y for some T-closed set F.
A is U-closed  Y  A is U-open.

Then Y  A can be expressed as:
Y  A = G  Y for some T-open set G.
From which A = Y  G  Y = X  Y  G  Y
= (X  G)  Y
= F  Y, where F = X  G is a T-closed set.
This completes the proof.

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