scholarly journals General rings of functions

1975 ◽  
Vol 20 (3) ◽  
pp. 359-365 ◽  
Author(s):  
A. Sultan
Keyword(s):  
General Setting ◽  
Continuous Functions ◽  
Tychonoff Space ◽  
Zero Sets ◽  
Rings Of Continuous Functions ◽  
Hewitt Realcompactification ◽  
Tychonoff Spaces

Several authors have studied various types of rings of continuous functions on Tychonoff spaces and have used them to study various types of compactifications (See for example Hager (1969), Isbell (1958), Mrowka (1973), Steiner and Steiner (1970)). However many important results and properties pertaining the Stone-Čech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.

scholarly journals R-P-spaces and subrings of C(X)

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 319-328 ◽  
Author(s):  
Mehdi Parsinia
Keyword(s):  
Continuous Functions ◽  
Tychonoff Space ◽  
New Approach ◽  
Rings Of Continuous Functions ◽  
Intermediate Ring

A Tychonoff space X is called a P-space if Mp = Op for each p ? ?X. For a subring R of C(X), we call X an R-P-space, if Mp ? R = Op ? R for each p ? ?X. Various characterizations of R-P-spaces are investigated some of which follows from constructing the smallest invertible subring of C(X) in which R is embedded, S-1R R. Moreover, we study R-P-spaces when R is an intermediate ring or an intermediate C-ring. We follow a new approach to some results of [W. Murray, J. Sack and S. Watson, P-spaces and intermediate rings of continuous functions, Rocky Mount. J. Math., to appear]. Also, some algebraic characterizations of P-spaces via intermediate rings are given. Finally, we establish some characterizations of C(X) among intermediate C-rings which are of the form I + C*(X), where I is an ideal in C(X).

scholarly journals Bounded resolutions for spaces $$C_{p}(X)$$ and a characterization in terms of X

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
J. C. Ferrando ◽  
J. Ka̧kol ◽  
W. Śliwa
Keyword(s):  
Tychonoff Space ◽  
Cosmic Space ◽  
Tychonoff Spaces

AbstractAn internal characterization of the Arkhangel’skiĭ-Calbrix main theorem from [4] is obtained by showing that the space $$C_{p}(X)$$ C p ( X ) of continuous real-valued functions on a Tychonoff space X is K-analytic framed in $$\mathbb {R}^{X}$$ R X if and only if X admits a nice framing. This applies to show that a metrizable (or cosmic) space X is $$\sigma $$ σ -compact if and only if X has a nice framing. We analyse a few concepts which are useful while studying nice framings. For example, a class of Tychonoff spaces X containing strictly Lindelöf Čech-complete spaces is introduced for which a variant of Arkhangel’skiĭ-Calbrix theorem for $$\sigma $$ σ -boundedness of X is shown.

scholarly journals A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Juan Carlos Ferrando
Keyword(s):  
Continuous Functions ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Bounded Sets

We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

scholarly journals Near-rings of continuous functions on disconnected groups

1979 ◽  
Vol 28 (4) ◽  
pp. 433-451 ◽  
Author(s):  
Robert D. Hofer
Keyword(s):  
Topological Group ◽  
Identity Element ◽  
Continuous Functions ◽  
Ideal Structure ◽  
Rings Of Continuous Functions ◽  
The Ideal

AbstractN(G) denotes the near-ring of all continuous selfmaps of the topological group G (under composition and the pointwise induced operation) and N0(G) is the subnear-ring of N(G) consisting of all functions having the identity element of G fixed. It is known that if G is discrete then (a) N0(G) is simple and (b) N(G) is simple if and only if G is not of order 2. We begin a study of the ideal structure of these near-rings when G is a disconnected group.

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