An example of a space whose all continuous mappings are almost injective
Iturralde, Pablo Mendoza
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001), p. 535-544 / Harvested from Czech Digital Mathematics Library
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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_\delta $-subsets of $X$ is finite. We also prove under CH that there exists a dense subspace $X$ of the real line $\Bbb R$, 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.

Publié le : 2001-01-01
Classification:  54C10,  54D18,  54D20,  54D30,  54E52 
@article{119268,
     author = {Pablo Mendoza Iturralde},
     title = {An example of a space whose all continuous mappings are almost injective},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {42},
     year = {2001},
     pages = {535-544},
     zbl = {1053.54022},
     mrnumber = {1860242},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119268}
}

Iturralde, Pablo Mendoza. An example of a space whose all continuous mappings are almost injective. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 535-544. http://gdmltest.u-ga.fr/item/119268/

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