# Baire category properties of function spaces with the Fell hypograph topology

## TL;DR

This paper investigates the Baire category properties of function spaces with the Fell hypograph topology, providing characterizations for various topological and category-related properties based on the spaces involved.

## Contribution

It offers a comprehensive characterization of when the function space is meager, Baire, or Polish, depending on the properties of the domain and codomain spaces.

## Key findings

- Characterizes pairs (X,Y) for which the function space is Baire or meager.
- Identifies conditions for the function space to be (almost) Polish or complete-metrizable.
- Provides criteria for the function space to have Choquet or strong Choquet properties.

## Abstract

For a Tychonoff space $X$ and a subspace $Y\subset\mathbb R$, we study Baire category properties of the space $C_{\downarrow F}(X,Y)$ of continuous functions from $X$ to $Y$, endowed with the Fell hypograph topology. We characterize pairs $X,Y$ for which the function space $C_{\downarrow F}(X,Y)$ is $\infty$-meager, meager, Baire, Choquet, strong Choquet, (almost) complete-metrizable or (almost) Polish.