# Proximity inductive dimension and Brouwer dimension agree on compact Hausdorff spaces

## TL;DR

This paper proves that the proximity inductive dimension and Brouwer dimension are equivalent on compact Hausdorff spaces, clarifying their relationship and implications for dimension theory.

## Contribution

It establishes the equivalence of proximity inductive and Brouwer dimensions on compact Hausdorff spaces, linking two important dimension concepts.

## Key findings

- Proximity inductive dimension agrees with Brouwer dimension on compact Hausdorff spaces.
- Fedorchuk's example shows a space where Brouwer dimension exceeds Lebesgue covering dimension.
- The example also demonstrates a discrepancy between proximity inductive and proximity dimensions.

## Abstract

In this paper we show that the proximity inductive dimension defined by Isbell agrees with the Brouwer dimension originally described by Brouwer on the class of compact Hausdorff spaces. Consequently, Fedorchuk's example of a compact Hausdorff space whose Brouwer dimension exceeds its Lebesgue covering dimension is an example of a space whose proximity inductive dimension exceeds its proximity dimension as defined by Smirnov.