An example of a space whose all continuous mappings are almost injective

Pablo Mendoza Iturralde

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 3, page 535-544
  • ISSN: 0010-2628

Abstract

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We show that all continuous maps of a space X onto second countable spaces are pseudo-open if and only if every discrete family of nonempty G δ -subsets of X is finite. We also prove under CH that there exists a dense subspace X of the real line , such that every continuous map of X is almost injective and X cannot be represented as K Y , where K is compact and Y is countable. This partially answers a question of V.V. Tkachuk in [Tk]. We show that for a compact X , all continuous maps of X onto second countable spaces are almost injective if and only if it is scattered. We give an example of a non-compact space Z such that every continuous map of Z onto a second countable space is almost injective but Z is not scattered.

How to cite

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Iturralde, Pablo Mendoza. "An example of a space whose all continuous mappings are almost injective." Commentationes Mathematicae Universitatis Carolinae 42.3 (2001): 535-544. <http://eudml.org/doc/248788>.

@article{Iturralde2001,
abstract = {We show that all continuous maps of a space $X$ onto second countable spaces are pseudo-open if and only if every discrete family of nonempty $G_\delta $-subsets of $X$ is finite. We also prove under CH that there exists a dense subspace $X$ of the real line $\mathbb \{R\}$, such that every continuous map of $X$ is almost injective and $X$ cannot be represented as $K\cup Y$, where $K$ is compact and $Y$ is countable. This partially answers a question of V.V. Tkachuk in [Tk]. We show that for a compact $X$, all continuous maps of $X$ onto second countable spaces are almost injective if and only if it is scattered. We give an example of a non-compact space $Z$ such that every continuous map of $Z$ onto a second countable space is almost injective but $Z$ is not scattered.},
author = {Iturralde, Pablo Mendoza},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {almost compact map; pseudo-open map; almost injective map; discrete family; scattered; almost injectivity; scattered space},
language = {eng},
number = {3},
pages = {535-544},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {An example of a space whose all continuous mappings are almost injective},
url = {http://eudml.org/doc/248788},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Iturralde, Pablo Mendoza
TI - An example of a space whose all continuous mappings are almost injective
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 3
SP - 535
EP - 544
AB - We show that all continuous maps of a space $X$ onto second countable spaces are pseudo-open if and only if every discrete family of nonempty $G_\delta $-subsets of $X$ is finite. We also prove under CH that there exists a dense subspace $X$ of the real line $\mathbb {R}$, such that every continuous map of $X$ is almost injective and $X$ cannot be represented as $K\cup Y$, where $K$ is compact and $Y$ is countable. This partially answers a question of V.V. Tkachuk in [Tk]. We show that for a compact $X$, all continuous maps of $X$ onto second countable spaces are almost injective if and only if it is scattered. We give an example of a non-compact space $Z$ such that every continuous map of $Z$ onto a second countable space is almost injective but $Z$ is not scattered.
LA - eng
KW - almost compact map; pseudo-open map; almost injective map; discrete family; scattered; almost injectivity; scattered space
UR - http://eudml.org/doc/248788
ER -

References

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  7. Tkachuk V.V., Spaces that are projective with respect to classes of mappings, Trans. Moscow Math. Soc. 50 (1988), 139-156. (1988) Zbl0662.54007MR0912056
  8. Tkachuk V.V., Remainders over discretes-some applications, Vestnik Moskov. Univ. Matematika 45 (1990), 4 18-21. (1990) MR1086601
  9. Tkačenko M.G., The Souslin property in free topological groups over compact spaces, Math. Notes 34 (1983), 790-793. (1983) MR0722229
  10. Uspenskii V.V., On spectrum of frequencies of function spaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 37 (1982), 01 31-35. (1982) MR0650600

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