Let be a completely regular space. Let be the StoneCech compactification of . In a previous post, we show that among all compactifcations of , the StoneCech compactification is maximal with respect to a partial order (see Theorem C2 in Two Characterizations of StoneCech Compactification). As a result of the maximality, is the largest among all compactifications of both in terms of cardinality and weight. We also establish an upper bound for the cardinality of and an upper bound for the weight of . As a result, we have upper bounds for cardinalities and weights for all compactifications of . We prove the following points.
 .
 .
 For every compactification of the space , .
 For every compactification of the space , .
 For every compactification of the space , .
 For every compactification of the space , .
Upper Bounds for StoneCech Compactification
StoneCech Compactification is Maximal
Upper Bounds for all Compactifications
It is clear that Results 5 and 6 follow from the preceding results. The links for other posts on StoneCech compactification can be found toward the end of this post
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Some Cardinal Functions
Let be a space. The density of is denoted by and is defined to be the smallest cardinality of a dense set in . For example, if is separable, then . The weight of the space is denoted by and is defined to be the smallest cardinality of a base of the space . For example, if is second countable (i.e. having a countable space), then . Both and are cardinal functions that are commonly used in topological discussion. Most authors require that cardinal functions only take on infinite cardinals. We also adopt this convention here. We use to denote the cardinality of the continuum (the cardinality of the real line ).
If is a cardinal number, then refers to the cardinal number that is the cardinallity of the set of all functions from to . Equivalently, is also the cardinality of the power set of (i.e. the set of all subsets of ). If (the first infinite ordinal), then is the cardinality of the continuum.
If is separable, then (as noted above) and we have and . Result 5 and Result 6 imply that is an upper bound for the cardinality of all compactifications of any separable space and is an upper bound of the weight of all compactifications of any separable space .
In general, Result 5 and Result 6 indicate that the density of bounds the cardinality of any compactification of by two exponents and the density of bounds the weight of any compactification of by one exponent.
Another cardinal function related to weight is that of the network weight. A collection of subsets of the space is said to be a network for if for each point and for each open subset of with , there is some set with . Note that sets in a network do not have to be open. However, any base for a topology is a network. The network weight of the space is denoted by and is defined to be the least cardinality of a network for . Since any base is a network, we have . It is also clear that for any space . Our interest in network and network weight is to facilitate the discussion of Lemma 2 below. It is a well known fact that in a compact space, the weight and the network weight are the same (see Result 5 in Spaces With Countable Network).
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Some Basic Facts
We need the following two basic results.

Lemma 1
Let be a space. Let be the set of all continuous functions . Then .

Lemma 2
Let be a space and let be a compact space. Suppose that is the continuous image of . Then .
Proof of Lemma 1
Let be a dense set with . Let be the set of all functions from to . Consider the map by . This is a onetoone map since whenever and agree on a dense set. Thus we have . Upon doing some cardinal arithmetic, we have . Thus Lemma 1 is established.
Proof of Lemma 2
Let be a continuous function from onto . Let be a base for such that . Let be the set of all where . Note that is a network for (since is a continuous function). So we have . Since is compact, (see Result 5 in Spaces With Countable Network). Thus we have .
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Results 1 and 2
Let be a completely regular space. Let be the unit interval . We show that the StoneCech compactification can be regarded as a subspace of the product space where (the product of many copies of ). The cardinality of is , thus leading to Result 1.
Let be the set of all continuous functions . The StoneCech compactification is constructed by embedding into the product space where each (see Embedding Completely Regular Spaces into a Cube or A Beginning Look at StoneCech Compactification). Thus is a subspace of where .
Note that . Thus can be regarded as a subspace of where . By Lemma 1, can be regarded as a subspace of the product space where .
To see Result 2, note that the weight of where is . Then , as a subspace of the product space, must have weight .
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Results 3 and 4
What drives Result 3 and Result 4 is the following theorem (established in Two Characterizations of StoneCech Compactification).

Theorem C2
Let be a completely regular space. Among all compactifications of the space , the StoneCech compactification of the space is maximal with respect to the partial order .
To define the partial order, for and , both compactifications of , we say that if there is a continuous function such that . See the following figure.
Figure 1
In this post, we use to denote this partial order as well as the order for cardinal numbers. Thus we need to rely on context to distinguish this partial order from the order for cardinal numbers.
Let be a compactification of . Theorem C2 indicates that (partial order), which means that there is a continuous such that (the same point in is mapped to itself by ). Note that is the image of under the function . Thus we have (cardinal number order). Thus Result 3 is established.
By Lemma 2, the existence of the continuous function implies that (cardinal number order). Thus Result 4 is established.
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Blog Posts on StoneCech Compactification

Post #0: Embedding Completely Regular Spaces into a Cube
Post #1: A Beginning Look at StoneCech Compactification
Post #2: Two Characterizations of StoneCech Compactification
Post #3: C*Embedding Property and StoneCech Compactification
Post #4 (this post): StoneCech Compactification is Maximal
Post #5: StoneCech Compactification of the Integers – Basic Facts
Post #6: StoneCech Compactifications – Another Two Characterizations
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Reference
 Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
 Willard, S., General Topology, AddisonWesley Publishing Company, 1970.
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