@article {IOPORT.05564232,
    author       = {Ershov, Yu.L.},
    title        = {On the classification of (effective) $\varphi $-spaces.},
    year         = {2009},
    journal      = {Annals of Pure and Applied Logic},
    volume       = {159},
    number       = {3},
    issn         = {0168-0072},
    pages        = {285-291},
    publisher    = {Elsevier Science B.V. (North-Holland), Amsterdam},
    doi          = {10.1016/j.apal.2008.07.007},
    abstract     = {The author studies the problem of classification of $\varphi$-spaces and of effective $\varphi$-spaces. All topological spaces are assumed to be $T_0$. The {\it specialization order} on such a space $X$ is defined as follows: $\xi\leqslant_X\eta\Leftrightarrow\xi\in cl_X[\eta]$. Let $\uparrow\xi=\{\eta\in X\mid \xi\leqslant_X\eta\}$. A space $X$ is called {\it $\varphi$-space} if the family of open sets of the kind $\{\uparrow\xi\mid \xi\in X\}$ forms its basis. The elements $\xi$ for which the set $\uparrow\xi$ is open are called {\it finitary}. Let $\Phi(X)$ be the set of all finitary elements of a space $X$. The paper contains a series of results related to classification of $\varphi$-spaces with the same space $\Phi(X)$ as well as some related results for effective presentations of such spaces. In particular, the author proves that, if $X_0$ is an $A$-discrete space, then the sobrification $S(X_0)$ of $X_0$ is a $\varphi$-space, and $\Phi(S(X_0))=X_0$, each intermediate space $X_0\subseteq Y\subseteq S(X_0)$ is a $\varphi$-space with $\Phi(Y)=X_0$, and all $\varphi$-spaces $Y$ with $\Phi(Y)=X_0$ are in some sense completely represented by such intermediate spaces. Further on, the author introduces and considers $u$-extensions. An extension $X\subseteq Y$ of topological spaces is called a $u$-extension if and only if for any continuous mapping $f:X\to Z$ into a topological space $Z$, there exists at most one continuous mapping $g:Y\to Z$ extending $f$. The author proves that an extension $X\subseteq Y$ is a $u$-extension if and only if $Y$ is homeomorphic to a subspace of $S(X)$ and that the sobrification $S(X)$ of every $\varphi$-space $X$ is a $\varphi$-space with $\Phi(X)=\Phi(S(X))$. Finally, the author defines the notion of effectivization of a $\varphi$-space and gives a characterization of effective $\varphi$-spaces with a fixed effectivization of the set of finitary elements.},
    reviewer     = {Andrei S. Morozov (Novosibirsk)},
    identifier   = {05564232},
}