Quotient spaces determined by algebras of continuous functions

Quotient Spaces Determined by Algebras of Continuous Functions

We prove that if X is a locally compact σ-compact space then on its quotient, γ(X) say, determined by the algebra of all real valued bounded continuous functions on X, the quotient topology and the completely regular topology defined by this algebra are equal. It follows from this that if X is second countable locally compact then γ(X) is second countable locally compact Hausdorff if and only i...

Quotient BCI-algebras induced by pseudo-valuations

In this paper, we study pseudo-valuations on a BCI-algebra and obtain some related results. The relation between pseudo-valuations and ideals is investigated. We use a pseudo-metric induced by a pseudovaluation to introduce a congruence relation on a BCI-algebra. We define the quotient algebra induced by this relation and prove that it is also a BCI-algebra and study its properties.

Spaces of Continuous Functions

Let X be a completely regular topological space, B(X) the Banach space of real-valued bounded continuous functions on X, with the usual norm ||&|| =supa?£x|&(#)| • A subset GCB(X) is called completely regular (c.r.) over X if given any closed subset KQ.X and point XoÇzX — K, there exists a ô £ G such that &(#o) = |NI a n ( i sup^^is: \b(x)\ <||&||. A topological space X is completely regular in...

On Algebras of Continuous Functions

Let C denote the algebra of functions continuous on the unit circle. With norm ||/¡| =sup¡xi-i \f(X)\, C is a Banach algebra. Let A denote the set of all / in C which are boundary values of functions analytic in \z\ <1 and continuous in \z\ ál. A is then a closed subalgebra of C, and by known results A consists of those and only those/ in C for which /|X|-l/(X)XBdX = 0, «^0. In [l ] the questio...

Banach Spaces and Algebras of Continuous Functions1