Page 197 - DMTH503_TOPOLOGY

Unit 22: The Tychonoff Theorem

Notes
Example 1: Let X be a topological space and let  be a closed sub-base for X and let {B } be
i
its generated closed base i.e. the class of all finite union of members of  if every class of B ’s with
i
the finite intersection property (FIP) has a non-empty intersection then X is compact.
Solution: Under the given hypothesis, we shall prove that X is compact. In order to prove the
required result it is sufficient to show that every basic cover of X has a finite sub-cover.
Let {O} be any basic open cover of X. Then X = O .
j j
j
C
C
Now, { B } being an open base for X implies that each O is a union of certain B ’s and the
i j i
C
totality of all such B ’s that arise in this way is a basic open cover of X. By De-Morgan’s law, the
i
totality of corresponding B ’s has empty intersection and therefore by the given hupothesis this
i
totality does not have FIP. This implies that there exist finitely many B ’s, say,
i
n
B ,B  B n i such that B   .
1 i 2 i K i
K 1

Taking complements on both sides, we set
n
C
B  X. (By De-Morgan’s Law)
K i

K 1
C
C
For each B (K = 1, 2, …, n) we can find a O such that B  O .
K i K j K i K j
n
Thus X = O .
K j

K 1
Thus, we have shown that every basic open cover of X has a finite sub-cover.
Example 2: Let X be a non-empty set. Then every class {B} of subsets of X with the FIP is
j
contained in some maximal class with the FIP.
Solution: Let {B} be a class of subsets of X with the FIP and let P be the family of all classes of
j
subsets of X that contains {B} and have the FIP.
j
For any F , F  P, define F  F so that F  F .
     
Then (P, ) is a partially ordered set. Let  be any totally ordered subset of (P, ). Then, the union
of all classes in  has an upper bound for  in P.
Thus (P, ) is a partially ordered set in which every totally ordered subset has an upper bound.
Hence by Zern’s lemma, P possesses a maximal element i.e., there exist a class {B } of subsets of
K
X such that {B}  {B }, {B } has the FIP and any class of subsets of X which properly contains {B }
j K K K
does not have the FIP.
Tychonoff’s Theorem

Before proving Tychonoff’s theorem, we shall prove two important lemmas.
Lemma 1: Let X be a set; Let  be a collection of subsets of X having the finite intersection
property. Then there is a collection D of subsets of X such that D contains  and D has the finite
intersection property, and no collection of subsets of X that properly contains D has this property.

We often say that a collection D satisfying the conclusion of this theorem is maximal with
respect to the finite intersection property.

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