On metrizable subspaces and quotients of non-Archimedean spaces Cp(X,K)

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 114; no. 3
• Main Authors: Ka̧kol, Jerzy, Śliwa, Wiesław
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Continuity (mathematics); Convergence; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Original Paper; Quotients; Subspaces; Theoretical; Topology; Function space; 46E10; 54C35; Pseudocompact space; Efimov space; Metrizable quotient; 54E35; Quotient space; Closed subspace
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-020-00849-9

Let K be a non-trivially valued non-Archimedean complete field. Let ℓ ∞ ( N , K ) [ ℓ c ( N , K ) ; c 0 ( N , K ) ] be the space of all sequences in K that are bounded [relatively compact; convergent to 0] with the topology of pointwise convergence (i.e. with the topology induced from K N ). Let X be an infinite ultraregular space and let C p ( X , K ) be the space of all continuous functions from X to K endowed with the topology of pointwise convergence. It is easy to see that C p ( X , K ) is metrizable if and only if X is countable. We show that for any X [with an infinite compact subset] the space C p ( X , K ) has an infinite-dimensional [closed] metrizable subspace isomorphic to c 0 ( N , K ) . Next we prove that C p ( X , K ) has a quotient isomorphic to c 0 ( N , K ) if and only if it has a complemented subspace isomorphic to c 0 ( N , K ) . It follows that for any extremally disconnected compact space X the space C p ( X , K ) has no quotient isomorphic to the space c 0 ( N , K ) ; in particular, for any infinite discrete space D the space C p ( β D , K ) has no quotient isomorphic c 0 ( N , K ) . Finally we investigate the question for which X the space C p ( X , K ) has an infinite-dimensional metrizable quotient. We show that for any infinite discrete space D the space C p ( β D , K ) has an infinite-dimensional metrizable quotient isomorphic to some subspace ℓ c 0 ( N , K ) of K N . If K is locally compact then ℓ c 0 ( N , K ) = ℓ ∞ ( N , K ) . If | n 1 K | ≠ 1 for some n ∈ N , then ℓ c 0 ( N , K ) = ℓ c ( N , K ) . In particular, C p ( β D , Q q ) has a quotient isomorphic to ℓ ∞ ( N , Q q ) and C p ( β D , C q ) has a quotient isomorphic to ℓ c ( N , C q ) for any prime number q .