Page 184 - DMTH503_TOPOLOGY

Topology

Notes Thus  a continuous map.
f : X  [a, b] s.t. f(F ) = {a}, f(F ) = {b}
1 2
19.2 Summary

 Urysohn’s lemma is a lemma that states that a topological space is normal iff any two
disjoint closed subsets can be separated by a function.
 Urysohn’s lemma is sometimes called “the first non-trivial fact of point set topology.”
 Urysohn’s lemma: If A and B are disjoint closed sets in a normal space X, then there exists
a continuous function f : X  [0, 1] such that  a  A, f(a) = 0 and  b  B, f(b) = 1.

19.3 Keywords

Continuous map: A continuous map is a continuous function between two topological spaces.

Disjoint: A and B are disjoint if their intersection is the empty set.
Normal: A topological space X is a normal space if, given any disjoint closed sets E and F, there
are open neighbourhoods U of E and V of F that are also disjoint.

Separated sets: A and B are separated in X if each is disjoint from the other’s closure. The closures
themselves do not have to be disjoint from each other.

19.4 Review Questions

1. Prove that every continuous image of a separable space is separable.

2. (a) Prove that the set of all isolated points of a second countable space is countable.
(b) Show that any uncountable subset A of a second countable space contains at least
one point which is a limit point of A.

3. (a) Let f be a continuous mapping of a Hausdorff non-separable space (X, T) onto itself.
Prove that there exists a proper non-empty closed subset A of X such that f(A) = A.
(b) Is the above result true if (X, T) is separable?

4. Examine the proof of the Urysohn lemma, and show that for given r,

1
f (r)  U  U ,
q
p
p r q r


p, q rational.
5. Give a direct proof of the Urysohn lemma for a metric space (X, d) by setting
d(x,A)
f(x) =
d(x,A) d(x,B)

6. Show that every locally compact Hausdorff space is completely regular.
7. Let X be completely regular, let A and B be disjoint closed subsets of X. Show that if A is
compact, there is a continuous function f : X  [0, 1] such that f(A) = {0} and f(B) = {1}.

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