We introduce the strong Gelfand–Phillips property for locally convex spaces and give several characterizations of this property. characterize among admitting a stronger Banach space topology. If $$C_{\mathcal {T}}(X)$$ is continuous functions on Tychonoff X, endowed with topology $${\mathcal {T}}$$ between pointwise compact-open topology, then: (a) has iff X contains compact countable subspace ...

We discuss the existence of complete accumulation points of sequences in products of topological spaces. Then we collect and generalize many of the results proved in Parts I, II and IV. The present Part VI is complementary to Part V to the effect that here we deal, say, with uniformity, complete accumulation points and κ-(λ)-compactness, rather than with regularity, [λ, μ]compactness and κ-(λ, ...

Pointfree topology is, as the name suggests, a way of studying spaces without (mentioning) points. This idea is more natural than one might initially think. For example, when drawing a point on paper, we do no draw an actual point, but a collection of points somewhere near the desired one. We drew a “spot”, which can be reduced in size if that would be required to serve our purposes. Hence it m...

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–Čech compactification βX (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing seq...

$C_{p}\left ( X\right )$ is distinguished $\Leftrightarrow$ the strong dual $L_{\beta }\left barrelled bidual $M\left ) =\mathbb {R}^{X}$. So one may judge how nearly by is, and also near dense subspace to Baire space $\mathbb Being Baire-like, always fairly close {R}^{X}$ in that sense. But if not distinguished, we show codimension of uncountable, i.e., algebraically far from {R}^{X}$, moreove...

Let \(\sum (X)\) be the collection of subrings C(X) containing \(C^{*}(X)\), where X is a Tychonoff space. For any \(A(X)\,{\in }\, \sum there associated subset \(\upsilon _{A}(X)\) \(\beta X\) which an A-analogue Hewitt real compactification X. (X)\), let [A(X)] class all \(B(X)\,{\in such that _{A}(X)=\upsilon _{B}(X)\). We show for first countable non compact space X, contains at least \(2^{...