Page 15 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

Theorem 5: In a topological space, an arbitrary union of open sets is open and a finite intersection Notes
of open sets is open. Prove it.
Proof: Let (X, T) be a topological space

Let G  T  i  N
i
 n
Let G = G i , H = G i


i 1 i 1
We are to prove that G and H are open subsets of X. By definition of topology,

(i) G  T  i  N  G i  T  G  T
i i 1

(ii) G  T  i  N  G  G  T
i 1 2
G  G  T, G  T
1 2 3
 G  G  G  T
1 2 3
By induction, it follows that
n
G i = H  T

i 1
Hence proved.
Theorem 6: In a topological space (X, T), prove that an arbitrary intersection of closed sets is
closed and finite union of closed sets is closed.
Proof: Let (X, T) be a topological space,

Let F  X be closed  i  N
i
 n
Let H = F i , F = F i

i 1 i 1

We are to prove that F and H are closed sets F is closed  i  N
i
 X – F is open  i  N
i
 n
Also, we know, (X F ) and (X F ) are open sets


i
i

i 1 i 1

[ An arbitrary union of open sets is open and a finite intersection of open sets is open]
 
 X – F i and X – F are open sets (by De Morgan’s Law)

i 1 i 1 i

 n
 F i , F i are closed sets (by definition of closed sets)
i 1 i 1


i.e. H, F are closed sets.
Hence, proved.
Self Assessment
5. Give two examples of a proper non-empty subset of a topological space such that it is both
open and closed and prove your assertion.

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