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One of the main obstacle to study compactness in topological spaces via ideals was the definition of ideal convergence of subsequences as in the existing literature according to which subsequence of an ideal convergent sequence may fail to be ideal convergent with respect to same ideal. This obstacle has been get removed in this article and notions of I compactness as well as I star compactness of topological spaces have been introduced and studied to some extent. Involvement of I nonthin subsequences in the definition of I and I star compactness make them different from compactness even in metric spaces.
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We show that an ideal $mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of $mathcal{I}$-limit points is comeager and, in addition, every accumulation point of $x$ is also an $mathcal{I}$-limit point (that is, a limit of a subsequence $(x_{n_k})$ such that ${n_1,n_2,ldots,} otin mathcal{I}$). The analogous characterization holds also for $mathcal{I}$-cluster points.
We prove that the minimal left ideals of the superextension $lambda(Z)$ of the discrete group $Z$ of integers are metrizable topological semigroups, topologically isomorphic to minimal left ideals of the superextension $lambda(Z_2)$ of the compact group $Z_2$ of integer 2-adic numbers.
A {it weak selection} on $mathbb{R}$ is a function $f: [mathbb{R}]^2 to mathbb{R}$ such that $f({x,y}) in {x,y}$ for each ${x,y} in [mathbb{R}]^2$. In this article, we continue with the study (which was initiated in cite{ag}) of the outer measures $lambda_f$ on the real line $mathbb{R}$ defined by weak selections $f$. One of the main results is to show that $CH$ is equivalent to the existence of a weak selection $f$ for which: [ mathcal lambda_f(A)= begin{cases} 0 & text{if $|A| leq omega$,} infty & text{otherwise.} end{cases} ] Some conditions are given for a $sigma$-ideal of $mathbb{R}$ in order to be exactly the family $mathcal{N}_f$ of $lambda_f$-null subsets for some weak selection $f$. It is shown that there are $2^mathfrak{c}$ pairwise distinct ideals on $mathbb{R}$ of the form $mathcal{N}_f$, where $f$ is a weak selection. Also we prove that Martin Axiom implies the existence of a weak selection $f$ such that $mathcal{N}_f$ is exactly the $sigma$-ideal of meager subsets of $mathbb{R}$. Finally, we shall study pairs of weak selections which are almost equal but they have different families of $lambda_f$-measurable sets.
A proper ideal $I$ in a commutative ring with unity is called a $z^circ$-ideal if for each $a$ in $I$, the intersection of all minimal prime ideals in $R$ which contain $a$ is contained in $I$. For any totally ordered field $F$ and a completely $F$-regular topological space $X$, let $C(X,F)$ be the ring of all $F$-valued continuous functions on $X$ and $B(X,F)$ the aggregate of all those functions which are bounded over $X$. An explicit formula for all the $z^circ$-ideals in $A(X,F)$ in terms of ideals of closed sets in $X$ is given. It turns out that an intermediate ring $A(X,F) eq C(X,F)$ is never regular in the sense of Von-Neumann. This property further characterizes $C(X,F)$ amongst the intermediate rings within the class of $P_F$-spaces $X$. It is also realized that $X$ is an almost $P_F$-space if and only if each maximal ideal in $C(X,F)$ is $z^circ$-ideal. Incidentally this property also characterizes $C(X,F)$ amongst the intermediate rings within the family of almost $P_F$-spaces.
For any ideal $mathcal{P}$ of closed sets in $X$, let $C_mathcal{P}(X)$ be the family of those functions in $C(X)$ whose support lie on $mathcal{P}$. Further let $C^mathcal{P}_infty(X)$ contain precisely those functions $f$ in $C(X)$ for which for each $epsilon >0, {xin X: lvert f(x)rvertgeq epsilon}$ is a member of $mathcal{P}$. Let $upsilon_{C_{mathcal{P}}}X$ stand for the set of all those points $p$ in $beta X$ at which the stone extension $f^*$ for each $f$ in $C_mathcal{P}(X)$ is real valued. We show that each realcompact space lying between $X$ and $beta X$ is of the form $upsilon_{C_mathcal{P}}X$ if and only if $X$ is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally-$mathcal{P}/$ almost locally-$mathcal{P}$, becomes a space of the same form. We further show that $C_mathcal{P}(X)$ is a free ideal ( essential ideal ) of $C(X)$ if and only if $C^mathcal{P}_infty(X)$ is a free ideal ( respectively essential ideal ) of $C^*(X)+C^mathcal{P}_infty(X)$ when and only when $X$ is locally-$mathcal{P}$ ( almost locally-$mathcal{P}$). We address the problem, when does $C_mathcal{P}(X)/C^mathcal{P}_{infty}(X)$ become identical to the socle of the ring $C(X)$. Finally we observe that the ideals of the form $C_mathcal{P}(X)$ of $C(X)$ are no other than the $z^circ$-ideals of $C(X)$.
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