# A note on covers defining relative and sectional categories

## TL;DR

This paper characterizes the relative category using open covers and extends the definitions of relative and sectional categories to arbitrary covers in ANR spaces, broadening their applicability.

## Contribution

It provides a new characterization of relative category via open covers and generalizes the definitions to arbitrary covers for ANR spaces.

## Key findings

- Relative category characterized by open covers
- Extension of sectional and relative categories to arbitrary covers in ANR spaces
- Broader applicability of categorical concepts in topological spaces

## Abstract

We give a characterization of relative category in the sense of Doeraene-El Haouari by using open covers. We also prove that relative category and sectional category can be defined by taking arbitrary covers (not necessarily open) when dealing with ANR spaces.