A number of properties enjoyed by two specially constructed topologies on $C(X)$

If $I$ is an ideal in the ring $C(X)$ of all real valued continuous functions defined over a Tychonoff space $X$, then $X$ is called $I$-$pseudocompact$ if the set $X\setminus \bigcap Z[I]$ is a bounded subset of $X$. Corresponding to $I$, the $m^I$-topology and $u^I$-topology on $C(X)$, generalizing the well-known $m$-topology and $u$-topology in $C(X)$ respectively are already there in the literature. It is proved amongst others that the $m^I$-topology is first countable if and only if the $u^I$-topology= $m^I$-topology on $C(X)$ if and only if $X$ is $I$-$pseudocompact$. A special case of this result on choosing $I=C(X)$ reads: the $u$-topology and $m$-topology on $C(X)$ coincide if and only if $X$ is pseudocompact. It is established that the $m^I$-topology on $C(X)$ is second countable if and only if it is $\aleph_0$-$bounded$ if and only if $X$ is compact, metrizable and $I=C(X)$. Furthermore it is realized that the $m^I$ topology on $C(X)$ is hemicompact if and only if it is $σ$-compact if and only if this topology is $H$-$bounded$ if and only if $X$ is finite and $I=C(X)$.

💡 Research Summary

The paper investigates two families of topologies on the ring C(X) of real‑valued continuous functions on a Tychonoff space X, obtained by fixing an ideal I⊂C(X). The classical uniform (u‑) topology and the pointwise (m‑) topology are generalized to the uⁱ‑ and mⁱ‑topologies respectively. For f∈C(X), ε>0 and u∈C⁺(X) (strictly positive continuous functions) the basic neighbourhoods are

 Bᵤ(f,I,ε)= {g∈C(X): supₓ|f(x)−g(x)|<ε and f−g∈I}

 Bₘ(f,I,u)= {g∈C(X): |f(x)−g(x)|<u(x) for all x and f−g∈I}.

The paper introduces the notion of I‑pseudocompactness: X is I‑pseudocompact if the set X\⋂_{g∈I}Z(g) is bounded (i.e., every real‑valued continuous function is bounded on it). When I=C(X) this reduces to ordinary pseudocompactness.

The main results can be grouped as follows:

1. First‑countability and coincidence of the two topologies.
   Theorem 2.2 proves the equivalence of three statements:
   (a) The mⁱ‑topology on C(X) is first countable;
   (b) The uⁱ‑ and mⁱ‑topologies coincide;
   (c) X is I‑pseudocompact.
   Earlier work required I to be convex; the present proof removes that restriction, showing that convexity is unnecessary. Consequently, the first‑countability of the finer mⁱ‑topology exactly characterises I‑pseudocompactness of the underlying space.

2. Second‑countability (ℵ₀‑boundedness) and compact metrizability.
   Theorem 4.3 establishes that the mⁱ‑topology is second countable (equivalently ℵ₀‑bounded) if and only if X is compact, metrizable, and I=C(X). The ℵ₀‑boundedness condition means that for every neighbourhood V of the identity there exists a countable set A with X=A·V. The proof shows that such a global countable base can exist only when the ideal is the whole ring, forcing the underlying space to be compact metric.

3. Hemicompactness, σ‑compactness and H‑boundedness.
   In Section 5 the authors prove that the following are equivalent:
   – The mⁱ‑topology is hemicompact;
   – It is σ‑compact;
   – It is H‑bounded (every sequence of neighbourhoods of the identity eventually covers each point).
   Moreover, these conditions hold precisely when X is a finite set and I=C(X). The H‑boundedness condition is very strong; the paper shows that it forces the function space to be essentially discrete.

4. Cardinal invariants.
   The character, weight, density, cellularity and Lindelöf number of Cₘⁱ(X) are all shown to coincide with the dominating number of the set X\⋂_{g∈I}Z(g). When I=C(X) this recovers known results for the classical m‑topology (e.g., Theorem 3.1 of