Realcompact and Lindelöf Extensions with Compact Remainder

Autor: Sosan Afrooz, Alireza Salehi, Abdolaziz Hesari
Rok vydání: 2019
Předmět:
Zdroj: Bulletin of the Iranian Mathematical Society. 46:1223-1242
ISSN: 1735-8515
1017-060X
Popis: An extension of a space X is a space Y which contains X as a dense subspace; $$Y{\setminus } X$$ is called the remainder of Y. In this paper, for a locally realcompact (resp., locally Lindelof) space X, we construct all realcompact (resp., Lindelof) extensions of X with compact remainder by means of some particular free ideals of C(X), the ring of all real-valued continuous functions on X. In fact, we are going to give a new approach different from that of similar works in the literature. For an ideal I of C(X), let $$\upsilon _IX$$ be the largest subspace of $$\beta X$$ on which every function in I can be extended continuously. We observe that every realcompact (resp., Lindelof) extension Y of X with compact remainder is precisely of the form $${\bar{i}}(\upsilon _IX)$$ , for a free ideal I contained in the ideal $$C_\mathbb {R}(X)$$ (resp., $$C_L(X)$$ ) consisting of all functions in C(X) with realcompact (resp., Lindelof) cozero-sets, where $${\bar{i}}:\beta X\rightarrow \beta Y$$ is the Stone extension of the identity mapping $$i:X\rightarrow X\subseteq \beta Y$$ . It turns out that locally realcompact (resp., locally Lindelof) spaces are the only spaces which have realcompact (resp., Lindelof) extensions with compact remainder. We conclude by constructing the largest (with respect to the standard partial order) one-point realcompact (resp., Lindelof) extension.
Databáze: OpenAIRE
Externí odkaz: