Geodesic
On topological classification of non-archimedean Fréchet spaces
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 2, pp. 457-463
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We prove that any infinite-dimensional non-archimedean Fréchet space $E$ is homeomorphic to $D^{\mathbb{N}}$ where $D$ is a discrete space with $\mathop {\mathrm card}(D)=\mathop {\mathrm dens}(E)$. It follows that infinite-dimensional non-archimedean Fréchet spaces $E$ and $F$ are homeomorphic if and only if $\mathop {\mathrm dens}(E)= \mathop {\mathrm dens}(F)$. In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field $\mathbb{K}$ is homeomorphic to the non-archimedean Fréchet space $\mathbb{K}^{\mathbb{N}}$.
We prove that any infinite-dimensional non-archimedean Fréchet space $E$ is homeomorphic to $D^{\mathbb{N}}$ where $D$ is a discrete space with $\mathop {\mathrm card}(D)=\mathop {\mathrm dens}(E)$. It follows that infinite-dimensional non-archimedean Fréchet spaces $E$ and $F$ are homeomorphic if and only if $\mathop {\mathrm dens}(E)= \mathop {\mathrm dens}(F)$. In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field $\mathbb{K}$ is homeomorphic to the non-archimedean Fréchet space $\mathbb{K}^{\mathbb{N}}$.
Classification : 46A04, 46S10
Keywords: non-archimedean Fréchet spaces; homeomorphisms 
@article{CMJ_2004_54_2_a17,
     author = {\'Sliwa, Wies{\l}aw},
     title = {On topological classification of non-archimedean {Fr\'echet} spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {457--463},
     year = {2004},
     volume = {54},
     number = {2},
     mrnumber = {2059266},
     zbl = {1080.46525},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2004_54_2_a17/}
}
TY  - JOUR
AU  - Śliwa, Wiesław
TI  - On topological classification of non-archimedean Fréchet spaces
JO  - Czechoslovak Mathematical Journal
PY  - 2004
SP  - 457
EP  - 463
VL  - 54
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2004_54_2_a17/
LA  - en
ID  - CMJ_2004_54_2_a17
ER  - 
%0 Journal Article
%A Śliwa, Wiesław
%T On topological classification of non-archimedean Fréchet spaces
%J Czechoslovak Mathematical Journal
%D 2004
%P 457-463
%V 54
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2004_54_2_a17/
%G en
%F CMJ_2004_54_2_a17
Śliwa, Wiesław. On topological classification of non-archimedean Fréchet spaces. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 2, pp. 457-463. http://geodesic.mathdoc.fr/item/CMJ_2004_54_2_a17/

[1] L. E. J. Brouver: On the structure of perfect sets of points. Proc. Acad. Amsterdam 12 (1910), 785–794.

[2] J. Kąkol, C. Perez-Garcia and W. Schikhof: Cardinality and Mackey topologies of non-Archimedean Banach and Fréchet spaces. Bull. Polish Acad. Sci. Math. 44 (1996), 131–141. | MR

[3] J. B.  Prolla: Topics in Functional Analysis over Valued Division Rings. North-Holland Math. Studies  77, North-Holland Publ. Co., Amsterdam, 1982. | MR | Zbl

[4] A. C. M. van Rooij: Notes on $p$-adic Banach spaces. Report 7633, Mathematisch Instituut, Katholieke Universiteit, Nijmegen, The Netherlands, 1976, pp. 1–62.

[5] A C. M. van Rooij: Non-Archimedean Functional Analysis. Marcel Dekker, New York, 1978. | MR | Zbl

[6] W. H.  Schikhof: Locally convex spaces over non-spherically complete valued fields. Bull. Soc. Math. Belgique 38 (1986), 187–207. | MR

[7] W.  Śliwa: Examples of non-archimedean nuclear Fréchet spaces without a Schauder basis. Indag. Math. (N.S.) 11 (2000), 607–616. | DOI | MR