"Domination by Second Countable Spaces" by Bernardo Cascales
 

Document Type

Lecture

Publication Date

2-25-2010

Abstract

Let �� be a compact space and let Δ = {(��,��) : �� ∈ ��} ⊆ �� × �� be its diagonal. Taking as a reference the known fact that �� is metrizable if and only if the complement of the diagonal (�� × ��) ∖ Δ is an ��σ in �� × ��, we move to the more intriguing case when (�� × ��) ∖ Δ = ⋃{��_α : α ∈ ℕ^ℕ}, where each ��_α is compact and ��_α ⊆ ��_β whenever α ≤ β. We prove that this assumption also implies metrizability when either {��_α : α ∈ ℕ^ℕ} is a fundamental family of compact subsets for (�� × ��) ∖ Δ or when MA(ω₁) is assumed. The success in proving these results relies upon the generation of usco maps: if we want to say it this way, it relies on some sort of understanding of compactoid filters. We provide applications (old and new) of the results and techniques presented here to functional analysis: metrizability of compact subsets in inductive limits, Lindelöf property of WCG Banach spaces, separability of Fréchet-Montel spaces, Lindelöf-σ character of spaces ��ₚ(��), etc. Students are welcome.

Relational Format

presentation

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.