Let be a completely regular space. Let be the Stone-Cech compactification of . We present two characterizations of in addition to three others that are discussed previously. In all, these five characterizations can help us derive many of the basic properties of . We prove the following theorems.
Let be a completely regular space. Every two completely separated subsets of have disjoint closures in .
The property described in Theorem C4 is unique to . That is, if is a compactification of satisfying the condition that every two completely separated subsets of have disjoint closures in , then must be .
Let be a normal space. Then every two disjoint closed subsets of have disjoint closures in .
If is a compactification of satisfying the property that every two disjoint closed subsets of have disjoint closures in , then is normal and must be .
The C theorem and U theorem with the same number work as a pair. The C theorem asserts that has a certain property. The corresponding U theorem asserts that of all the compactifications of , is the only one with the property in question. Whenever we can show a given compactification does not possess the property described in the C-U theorem pair, we know that that compactification is not (consequence of the C theorem). Whenever we can show that a given compactification has the property described in the C-U theorem pair, we know that that compactification must be (a consequence of the U theorem).
Three other sets of characterizations (Theorems C1, U1, C2, U2, C3 and U3) have been established previously. See the links found below.
Completely Separated Sets
Let be a completely regular space. Let and . The sets and are said to be completely separated in if there is a continuous function such that for each , and for each , (this can also be expressed as and ). If and are completely separated, and are necessarily disjoint closed sets, since and .
The Urysohn’s lemma can be stated as: a space is a normal space if and only if every two disjoint closed sets are completely separated. Thus disjoint closed sets are not necessarily completely separated (such sets can be found in non-normal spaces).
Some Helpful Results
To prove Theorem U4, we need a lemma and a theorem. Most of the work in proving Theorem U4 is carried out in Theorem 2 below.
Let be a compact space. Let be an open subset of . Let be a collection of compact subsets of such that . Then there exists a finite collection such that .
Proof of Lemma 1
Let , which is compact. Let be the collection of all where . Note that implies that . Thus is a collection of open sets covering the compact set . We have such that . Each for some . Now is the desired finite collection.
Let be a completely regular space. Let be a dense subspace of . Let be a continuous function from into a compact space . Suppose that every two completely separated subsets of have disjoint closures in . Then can be extended to a continuous .
For each , let be the set of all open subsets of containing . For each , let be the set of all where . Note that each consists of compact subsets of . The theorem is established by proving the following claims.
For each , the collection has non-empty intersection.
For any , we have the following:
The above shows that has the finite intersection property (f. i. p.). It is a well known fact that in a compact space, any collection of sets with f. i. p. has non-empty intersection (see  or  or see The Finite Intersection Property in Compact Spaces and Countably Compact Spaces in this blog).
For each , has only one point.
Let . Suppose that
Then there exist open subsets and of such that , and . Since is compact, it is a normal space. By the Urysohn’s lemma, there exists a continuous such that for each , and for each , . Then because of the function , the sets and are completely separated sets in . By assumption, these two sets have disjoint closures in , i.e.,
The point cannot be in both of the sets in . Assume the following:
Then . Note that . Furthermore, . Thus we have:
Since and is an open set containing , contains at least one point of . Choose such that . Now choose such that . First we have and thus . Secondly since , we have . We now have and , a contradiction. If we assume , we can also derive a contradiction in a similar derivation. Thus the assumption in above is faulty. The intersection can only have one point.
For each , .
Let . Suppose that where . the rest of the proof for Claim 3 is similar to that of Claim 2. For the sake of completeness, we give a sketch.
There exist open subsets and of such that , and . By the same argument as in Claim 2, we have the condition , i.e., . Since , . The remainder of the proof of Claim 3 is the same as above starting with condition with . A contradiction will be obtained. We can conclude that the assumption that where must be faulty. Thus Claim 3 is established.
For each , define by letting be the point in . Note that this function extends . Furthermore, the map is continuous.
To show is continuous, let and let where is open in . The collection is a collection of compact subsets of such that . By Lemma 1, there exists such that . By the definition of , there exists such that each . Let . We have:
Note that is an open subset of and . We show that . Pick . According to the definition of , we have . Since , we have . Thus by , we have . Thus Claim 4 is established.
With all the above claims established, we completed the proof of Theorem 2.
Theorem C4 and Theorem U4
Proof of Theorem C4
In proving C4, we use Theorem C3, which is found in C*-Embedding Property and Stone-Cech Compactification.
Let and be two completely separated sets in . Then there exists some continuous such that for each , and for each , . By Theorem C3, is extended by some continuous . The sets and are disjoint closed sets in . Furthermore, and . Thus and have disjoint closures in .
Proof of Theorem U4
In proving U4, we use Theorem U1, which is stated and proved in Two Characterizations of Stone-Cech Compactification.
Suppose that is a compactification of satisfying the condition that every two completely separated subsets of have disjoint closures in . Let be a continuous function from into a compact space . By Theorem 2, can be extended by a continuous . By Theorem U1, must be .
Theorem C5 and Theorem U5
Proof of Theorem C5
Let be a normal space. According to the Urysohn’s lemma, every two disjoint closed sets are completely separated. Thus by Theorem C4, every two disjoint closed subsets of have disjoint closures in .
Proof of Theorem U5
Suppose that is a compactification of satisfying the property that every two disjoint closed subsets of have disjoint closures in . To show that is normal, let and be disjoint closed subsets of . By assumption about , and (closures in ) are disjoint. Since are compact and Hausdorff, is normal. Then and can be separated by disjoint open subsets and of . Thus and are disjoint open subsets of separating and .
We use Theorem U4 to prove Theorem U5. We show that satisfies Theorem U4. To this end, let and be two completely separated sets in . We show that and have disjoint closures in . There exists some continuous such that for each , and for each , . Then and are disjoint closed sets in such that and . By assumption about , and have disjoint closures in . This implies that and have disjoint closures in . Then by Theorem U4, must be .
Blog Posts on Stone-Cech Compactification
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.