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Stone-Cech Compactifications of Infinite Discrete Spaces

1. Introducton.

This paper is a study of the nature of the Stone-Cech compacti-

fic~tion f3X of an infinitediscrete space X. If X is an infinite discrete space, then it is clear that X contains as many disjoint topological copies of itself as there are points in X. From this, we deduce that f3X also has thisproperty, that is, f3X contains as many disjoint copies of itself as there are points in f3X. Some sufficient conditions that subsets of f3X be countably compact are also given.
2. Preliminary concepts.Let Nand R denote the space of natural numbers and the space of real numbers, respectively. The cardinality of a set X is denoted by IXI, with the exception that we use ~o and c in place of INI andIRI, respectively. Let X be a space. The closure in X of a subset S of X i3 denoted by clxS. If X is a completely regular Tt-space we denote by C* (X) t he set of bounded continuous functions from Xinto R. A subset S of X is C* embedded in X if every function in C* (S) extends to a function in C*(X). Thus X is C io-embedded in the Stonc-Cech compactification f3X of X. It will be convenient torecall here that f3X is characterized by the property of being a compact Hausdorff space having X as a C*-embedded dense subset. From this, it follows immediately that a sub3ct S of X is C>··embedded in Xif and only if f3S is identical with the closure in f3X of S. We also note that the Tietze extension theorem is phrased as: Closed set:> are C*-embedded in a normal space.
3. Disjoint copies of fJXin fJX - X. As mentioned in the introduction, the purpose of this section is to prove that the Stone-Cech compactification of an infinite discrete space contains as many disjoint copies of itself a3its cardinality. To do this, the following lemma will be needed. LEMMA 1. If a regular space is the sum of a discrete open subset and a countable set then it is paracompact. Proof. Let X be a regular...
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