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
Recommended Citation
Cascales, Bernardo, "Domination by Second Countable Spaces" (2010). Analysis Seminar. 26.
https://egrove.olemiss.edu/math_analysis/26