We give an abstract version of Sierpiński's theorem which says that the closure in the uniform convergence topology of the algebra spanned by the sums of lower and upper semicontinuous functions is the class of all Baire 1 functions. Later we show that a natural generalization of Sierpiński's result for the uniform closure of the space of all sums of A and CA functions is not true. Namely we show that the uniform closure of the space of all sums of A and CA functions is a proper subclass of the...

We prove that there exists an example of a metrizable non-discrete space $X$, such that $C_p(X\times \omega ){\approx }_l{C}_p\left(X\right)$ but $C_p(X\times S)\neg {\approx }_l{C}_p\left(X\right)$ where $S=(\left\{0\right\}\cup \{\frac{1}{n+1}:n\in \omega \})$ and $C_p\left(X\right)$ is the space of all continuous functions from $X$ into reals equipped with the topology of pointwise convergence. It answers a question of Arhangel’skii ([2, Problem 4]).

We present a simple proof of a Banach-Stone type Theorem. The method used in the proof also provides an answer to a conjecture of Cao, Reilly and Xiong.

This work provides an evaluating complete description of positive homomorphisms on an arbitrary algebra of real-valued functions.

In the present paper we give a duality between a special type of ideals of subalgebras of $C\left(X\right)$ containing $C^{*}\left(X\right)$ and $z$-filters of $\beta X$ by generalization of the notion $z$-ideal of $C\left(X\right)$. We also use it to establish some intersecting properties of prime ideals lying between $C^{*}\left(X\right)$ and $C\left(X\right)$. For instance we may mention that such an ideal becomes prime if and only if it contains a prime ideal. Another interesting one is that for such an ideal the residue class ring is totally ordered if and only if it is prime.

Let $X$ be a locally connected, $b$-compact metric space and $E$ a closed subset of $X$. Let $𝔾$ be the space of all continuous real-valued functions defined on some closed subsets of $E$. We prove the equivalence of the ${\tau }_{{}_{aw}}$ and ${\tau }_{{}_K}^c$ topologies on $𝔾$, where ${\tau }_{{}_{aw}}$ is the so called Attouch-Wets topology, defined in terms of uniform convergence of distance functionals, and ${\tau }_{{}_K}^c$ is the topology of Kuratowski convergence on compacta.