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A net theoretic approach to homotopy theory

Renan Maneli Mezabarba, Rodrigo Santos Monteiro, Thales Fernando Vilamaior Paiva

Abstract

This paper uses a net-theoretic approach to convergence spaces, aimed to simplify the description of continuous convergence in order to apply it in problems concerning Homotopy Theory. We present methods for handling homotopies of limit spaces, define fundamental groupoids, and prove a generalized version of the Seifert-van Kampen Theorem for limit spaces.

A net theoretic approach to homotopy theory

Abstract

This paper uses a net-theoretic approach to convergence spaces, aimed to simplify the description of continuous convergence in order to apply it in problems concerning Homotopy Theory. We present methods for handling homotopies of limit spaces, define fundamental groupoids, and prove a generalized version of the Seifert-van Kampen Theorem for limit spaces.

Paper Structure

This paper contains 5 sections, 8 theorems, 13 equations, 1 figure.

Table of Contents

  1. Nets (and filters) and convergence spaces
  2. Convergential translations of topological notions
  3. Homotopy and fundamental groupoids in limit spaces
  4. The Seifert-Van Kampen Theorem for fundamental groupoids of limit spaces
  5. Further remarks and comments

Key Result

Proposition 1.1

For topological spaces $\left\langle X,\tau\right\rangle$ and $\left\langle Y,\tau'\right\rangle$, a function $f\colon X\to Y$ is continuous if and only if $f\circ \varphi$$\tau'$-converges to $f(x)$ for every $x\in X$ and every net $\varphi$ in $X$ such that $\varphi\to_\tau x$.

Figures (1)

  • Figure 1: The space $X$.

Theorems & Definitions (25)

  • Proposition 1.1: Folklore
  • proof
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1: cf. preuss
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.1
  • proof : Proof of $(3)$
  • ...and 15 more