Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Selection Games with Minimal Usco Maps

Christopher Caruvana

Abstract

We establish relationships between various topological selection games involving the space of minimal usco maps with various topologies, including the topology of pointwise convergence and the topology of uniform convergence on compact sets, and the underlying domain using full- and limited-information strategies. We also tie these relationships to analogous results related to spaces of continuous functions. The primary games we consider include Rothberger-like games, generalized point-open games, strong fan-tightness games, Tkachuk's closed discrete selection game, and Gruenhage's $W$-games.

Selection Games with Minimal Usco Maps

Abstract

We establish relationships between various topological selection games involving the space of minimal usco maps with various topologies, including the topology of pointwise convergence and the topology of uniform convergence on compact sets, and the underlying domain using full- and limited-information strategies. We also tie these relationships to analogous results related to spaces of continuous functions. The primary games we consider include Rothberger-like games, generalized point-open games, strong fan-tightness games, Tkachuk's closed discrete selection game, and Gruenhage's -games.
Paper Structure (7 sections, 28 theorems, 73 equations)

This paper contains 7 sections, 28 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Selection Games
  4. Uniform Spaces
  5. Usco Mappings
  6. Results
  7. Questions

Key Result

Lemma 8

For a space $X$ and an ideal of closed sets $\mathcal{A}$ of $X$, $\mathcal{O}_X(\mathcal{A}) = \Lambda_X(\mathcal{A})$.

Theorems & Definitions (68)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 8: See CHContinuousFunctions
  • Definition 9
  • Definition 10
  • ...and 58 more