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Better Subtopologies

Alexander Arhangel'skii, Raushan Buzyakova

TL;DR

This work addresses how to realize a space $X$ with property $A$ as a continuous bijective image of a space with property $B$, and when the concurrent presence of both properties can be ensured. The core method refines the target GO-space $Y$ to a first-countable GO-space $Z$ while preserving a continuous bijection from $X$, using a convex-open basis plus rays at points of uncountable character in the initial segments. A key result is that $Z$ can be constructed so that the map $X\to Z$ remains a continuous bijection, and if $Y$ is a subspace of an ordinal, $Z$ is a subordinal subspace; a counterexample based on the long segment shows the limits of replacing GO-spaces with LOTS. The paper also raises several open questions regarding countable pseudocharacter, ind-dimension, and possible inductive refinements, indicating directions for further study in ordered-space representations and factorization phenomena.

Abstract

We study conditions under which a space that has a good property and a courser topology with another good property admits a continuous bijection onto a space with both properties.

Better Subtopologies

TL;DR

This work addresses how to realize a space with property as a continuous bijective image of a space with property , and when the concurrent presence of both properties can be ensured. The core method refines the target GO-space to a first-countable GO-space while preserving a continuous bijection from , using a convex-open basis plus rays at points of uncountable character in the initial segments. A key result is that can be constructed so that the map remains a continuous bijection, and if is a subspace of an ordinal, is a subordinal subspace; a counterexample based on the long segment shows the limits of replacing GO-spaces with LOTS. The paper also raises several open questions regarding countable pseudocharacter, ind-dimension, and possible inductive refinements, indicating directions for further study in ordered-space representations and factorization phenomena.

Abstract

We study conditions under which a space that has a good property and a courser topology with another good property admits a continuous bijection onto a space with both properties.
Paper Structure (2 sections, 3 theorems)

This paper contains 2 sections, 3 theorems.

Table of Contents

  1. Introduction
  2. Study

Key Result

Theorem 2.1

Let $X$ be a first-countable space that admits a continuous bijection onto a GO-space $Y$. Then $X$ admits a continuous bijection onto a first-countable GO-space.

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Example 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.5
  • proof