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Spaces where all bijections are morphisms

Lucas H. R. de Souza

Abstract

Here we classify all topological spaces where all bijections to itself are homeomorphisms. As a consequence, we also classify all topological spaces where all maps to itself are continuous. Analogously, we classify all measurable spaces where all bijections to itself are measurable with measurable inverse. As a consequence, we also classify all measurable spaces where all maps to itself are measurable.

Spaces where all bijections are morphisms

Abstract

Here we classify all topological spaces where all bijections to itself are homeomorphisms. As a consequence, we also classify all topological spaces where all maps to itself are continuous. Analogously, we classify all measurable spaces where all bijections to itself are measurable with measurable inverse. As a consequence, we also classify all measurable spaces where all maps to itself are measurable.
Paper Structure (3 sections, 14 theorems)

This paper contains 3 sections, 14 theorems.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. Proof of Theorem 2

Key Result

Theorem 1

Let $X$ be a topological space. Then $Homeo(X) = bij(X)$ if and only if $X$ is cocardinal.

Theorems & Definitions (26)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Definition 2
  • Theorem 2
  • Corollary 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 16 more