The continuity properties of compact-preserving functions

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is compact-preserving if and only if for each point $x\in X$ there is a compact subset $K_x\subset Y$ such that for each neighborhood $O_{f(x)}\subset Y$ of $f(x)$ there is a neighborhood $O_x\subset X$ of $x$ such that $f(O_x)\subset O_{f(x)}\cup K_x$ and the set $K_x\setminus O_{f(x)}$ is finite. This characterization is applied to give an alternative proof of a classical characterization of continuous functions on locally connected metrizable spaces as functions that preserve compact and connected sets. Also we show that for each compact-preserving function $f:X\to Y$ defined on a (strong) Fr'echet space $X$, the restriction $f|LI’_f$ (resp. $f|LI_f)$ is continuous. Here $LI_f$ is the set of points $x\in X$ of local infinity of $f$ and $LI’_f$ is the set of non-isolated points of the set $LI_f$. Suitable examples show that the obtained results cannot be improved.

💡 Research Summary

The paper investigates functions that preserve compactness—i.e., functions $f\colon X\to Y$ such that for every compact subset $K\subset X$, the image $f(K)$ is compact in $Y$. While every continuous map is compact‑preserving, the converse fails in general. The authors focus on domains that are strong Fréchet spaces, a class of spaces where any point belonging to the closure of a set can be approached by a sequence drawn from that set. This sequential accessibility enables a fine‑grained analysis of compact‑preserving maps.

The central result is a characterization theorem: a function $f$ defined on a strong Fréchet space $X$ is compact‑preserving iff for each $x\in X$ there exists a compact subset $K_x\subset Y$ satisfying two conditions:

1. $f(x)\in K_x$;
2. For every neighbourhood $O_{f(x)}$ of $f(x)$ there is a neighbourhood $O_x$ of $x$ such that
   $$f(O_x)\subset O_{f(x)}\cup K_x,$$
   and the set $K_x\setminus O_{f(x)}$ is finite.

Thus $f$ behaves like a continuous map at $x$, except possibly on a finite exceptional set $K_x\setminus O_{f(x)}$. The proof proceeds in two directions. Assuming compact‑preservation, the authors construct $K_x$ by collecting all limit points of images of sequences converging to $x$; the strong Fréchet property guarantees that any failure of continuity can be captured by a finite subset of this compact set. Conversely, given such $K_x$ and neighbourhoods, any compact $K\subset X$ can be covered by finitely many $O_x$’s, and the corresponding images differ from a compact set by only finitely many points, which preserves compactness. Hence the condition is both necessary and sufficient.

Using this characterization, the authors give a new proof of a classical theorem: on a locally connected metrizable space $X$, a function $f\colon X\to Y$ is continuous precisely when it preserves both compact and connected subsets. The traditional proof relies on delicate arguments about the preservation of connectedness. Here, the authors first apply the characterization to obtain the “almost continuity” at each point. Local connectedness guarantees that the finite exceptional set cannot break the connectedness of small neighbourhoods, forcing $f$ to be genuinely continuous. This approach streamlines the argument and highlights the role of compactness as the primary driver of continuity under the additional hypothesis of local connectedness.

The paper further studies the behavior of compact‑preserving maps on special subsets of the domain. Define:

• $LI_f$ = { $x\in X$ | for every neighbourhood $U$ of $x$, the image $f(U)$ is infinite } (points of local infinity);
• $LI’_f$ = the set of non‑isolated points of $LI_f$.

The authors prove that for a compact‑preserving $f$ on a (strong) Fréchet space, the restrictions $f|{LI_f}$ and, more strongly, $f|{LI’_f}$ are continuous. The intuition is that at points of local infinity the exceptional compact set $K_x$ must be finite, and because $f(U)$ is infinite for every neighbourhood $U$, the only way to accommodate the finiteness condition is for $f$ to be continuous there. In the non‑isolated part $LI’_f$, the strong Fréchet property allows one to pass to convergent sequences, yielding a direct verification of continuity.

To demonstrate the sharpness of their results, the authors present several counterexamples:

1. Necessity of the strong Fréchet condition: on a space that is merely Fréchet (or not Fréchet at all), the characterization fails; a function may satisfy the finite‑exception condition locally but still not preserve compactness globally.
2. Finiteness of the exceptional set: if $K_x\setminus O_{f(x)}$ is allowed to be infinite, compact‑preservation can break down.
3. Role of $LI’_f$: restricting to $LI_f$ alone does not guarantee continuity when $LI_f$ contains isolated points; the restriction to $LI’_f$ is essential.

These examples confirm that each hypothesis in the main theorems is optimal and cannot be weakened without losing the conclusions.

In summary, the paper contributes a precise structural description of compact‑preserving functions on strong Fréchet spaces, revealing that such maps are “continuous up to finitely many points” at each location. This insight yields a streamlined proof of the classical compact‑and‑connected preservation characterization of continuity on locally connected metrizable spaces, and it uncovers new continuity phenomena on the sets of points where the image is locally infinite. The results deepen our understanding of the interplay between compactness, continuity, and sequential properties in topology, and they open avenues for further exploration of compact‑preserving maps under alternative topological constraints.