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.

**Theorem C4**

Let be a completely regular space. Every two completely separated subsets of have disjoint closures in .

**Theorem U4**

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 .

**Theorem C5**

Let be a normal space. Then every two disjoint closed subsets of have disjoint closures in .

**Theorem U5**

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.

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**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).

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**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.

**Lemma 1**

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.

**Theorem 2**

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 .

**Proof**

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.

**Claim 1**

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 [1] or [2] or see The Finite Intersection Property in Compact Spaces and Countably Compact Spaces in this blog).

**Claim 2**

For each , has only one point.

Let . Suppose that

where

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.

**Claim 3**

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.

**Claim 4**

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.

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**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 .

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**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 .

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**Blog Posts on Stone-Cech Compactification**

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**Reference**

- Engelking, R.,
*General Topology, Revised and Completed edition*, Heldermann Verlag, Berlin, 1989.
- Willard, S.,
*General Topology*, Addison-Wesley Publishing Company, 1970.

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