Nearnesses, proximities, and 𝑇₁-compactifications

Abstract

Abstract. Gagrat, Naimpally, and Thron together have shown that separated Lodato proximities yield 7",-compactifications, and conversely. This correspondence is not 1-1, since nonequivalent compactifications can induce the same proximity. Herrlich has shown that if the concept of proximity is replaced by that of nearness then all principal (or strict) 7",-extensions can be accounted for. (In general there are many nearnesses compatible with a given proximity.) In this paper we obtain a 1-1 correspondence between principal 7",-extensions and cluster-generated nearnesses. This specializes to a 1-1 match between principal Trcompactifications and contigua! nearnesses.These results are utilized to obtain a 1-1 correspondence between Lodato proximities and a subclass of T,-compactifications. Each proximity has a largest compatible nearness, which is contigua! The extension induced by this nearness is the construction of Gagrat and Naimpally and is characterized by the property that the dual of each clan converges. Hence we obtain a 1-1 match between Lodato proximities and clan-complete principal r,-compactifications. When restricted to ¿/-proximities, this correspondence yields the usual map between r2-compactifications and ¿/'-proximities.

Nearnesses and T0-Extensions of Topological Spaces

“…The collection {T(y):yEY} is called the trace system induced by K. Now suppose that X is a T[-space, and let v be a compatible, cluster generated, Lodato nearness on X. In [3], Reed shows how to construct a T[-extension K v of X, induced by v in a natural manner. The construction is similar to that of Bentley and Herrlich [1].…”

“…In [3], Reed shows that the maps K ~ V K and v ~ K v are inverses on the sets of (equivalence classes of) principal Tcextensions of a TJ-space X and the compatible, cluster-generated, Lodato nearnesses on X, thus obtaining the following lovely result: THEOREM 3 (Reed [3]). The map K ~ V K is a one-to-one correspondence between the principal TJ-extensions of a TJ-space X and the compatible, cluster generated, Lodato nearnesses on X.…”

“…It has since proven to be a useful tool in the classification of extensions of topological spaces. Reed [3], Bentley and Herrlich [1], and others have used nearnesses to classify the principal T1-extensions of a Trspace. In [3], Reed obtained a one to-one correspondence between the principal T1-extensions of a T1-space X and the compatible, cluster-generated Lodato nearnesses on X.…”

“…Reed [3], Bentley and Herrlich [1], and others have used nearnesses to classify the principal T1-extensions of a Trspace. In [3], Reed obtained a one to-one correspondence between the principal T1-extensions of a T1-space X and the compatible, cluster-generated Lodato nearnesses on X. In this paper, we look at the question of characterizing To-extensions using nearnesses, namely we generalize Reed's result to classify the principal To-extensions of a To-space X.…”

“…We begin by recalling some basic definitions concerning nearnesses, grills, and topological extensions. Throughout this paper, we use the notation of Reed [3].…”