On zero-dimensionality of remainders of some compactications.

Abstract

A compactication of a topological space is a dense embedding of the space into a compact topological space. We study dierent methods of compactifying a topological space with the focus on zero-dimensionality of the remainder. Freudenthal compactication is known as a maximal compactication with a zero-dimensional remainder and is guaranteed to exist for rim-compact spaces. It is shown that this compactication can be characterized using proximities. In fact, there is a one-to-one correspondence between compactications and proximities and, in particular, between compactications with zero-dimensional remainder and zero-dimensional proximity. Almost rim-compact spaces are spaces that are larger than the rim-compact spaces and they are shown to also have a compactication with a zero-dimensional remainder. But these do not exhaust spaces that have a compactication with a zero-dimensional remainder, for example, recently it was found that spaces that lie between the locally compact part and its Freudenthal compactication also have a zero-dimensional remainder. It is known that the Freudenthal compactication is also perfect, we study the relationship between maximum compactications with a zero-dimensional remainder and the perfectness of these compactications.

Iqoqa. Ukuhlangana kuyisici esibaluleke kakhulu setopology. Umbuzo wokuthi ngabe indawo yetopology ingenziwa icwecwe (ihlanganiswe) ngokwengeza amaphuzu athile kuyo indala kakhulu futhi iye yacatshangelwa ochwepheshe bezibalo abaningi. Iqoqo lamaphuzu engeziwe libizwa ngokuthi ingxenye esele yesikhala esikhona kulesi sikhala esisha esihlangene. IFreudenthal compactification iyona enendawo lapho okusele kuzero-dimensional. Lo mqingo wocwaningo ucwaninga ukuhlangana kwalolu hlobo lwetopology engenamaphuzu. Senza lokhu ngokwethula umbono wezero-dimensional embeddedness futhi sibonisa ukuthi lo mbono unamandla kunezero-dimensionality yakudala. Ukuminyanisa okusele okuzero-dimensionally embedded kuyacwaningwa. Kuvela ukuthi ukuhlanganisa okunjalo kuhlobene nokunye okuphelele. Sethula uhlobo lokuhlanganisa okungaphelele futhi sibonisa ukuthi lolu hlobo luqukethe konke okuphelele. Ukuhlangana okusele okuzero-dimensionally embedded kuphelele ngokunembile uma kuphelele. Sibonisa ukuthi insalela yohlaka ekuhlanganisweni kweFreudenthal inguzero-dimensionally embedded nokuthi uhlaka luhlangene emphethweni uma futhi kuphela inokuhlanganisa okusele okuzero-dimensionally embedded. Ubufakazi bokuthi konke ukuhlanganiswa kwe-N-star ezero-dimensionally embedded remainder kuyanikezwa. Kuyaziwa ukuthi ukuhlanganiswa kwe-Freudenthal kohlaka oluhlangene emphethweni kuwukuhlanganisa okuncane komugqa. Phakathi kwezinye izinto, lokhu kokugcina kusetshenziselwa ukukhombisa ukuthi iFreudenthal compactification iyona ndlela encane kakhulu yokuqoqwa kokuhlanganiswa kwe-N-star. Ubusha bemiphumela equkethwe lapha buseqinisweni lokuthi ubufakazi bayo ngokuvamile abuncikile kunoma yisiphi isimiso sokuzikhethela.