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

@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 -