A remark on universal marked spaces

Abstract

By {\it marked space} we mean a pair $(X,a^X)$, where $X$ is a
topological space and $a^X$ is a fixed point of $X$. By {\it
embedding of a marked space $(Y,a^Y)$ into a marked space $(X,a^X)$}
we mean an embedding $i$ of $Y$ into $X$ for which $i(a^Y)=a^X$. In
this paper it is proved that: $(1)$ in the class of all marked
spaces there are universal elements and $(b)$ in the Alexandroff
cube $F^\tau $ there are points $a$, such that the marked space
$(F^\tau,a)$ is universal in the class of all marked spaces (note
that not all points of $F^\tau $ satisfy this condition), which are
answers to the corresponding problems, putting in
\cite{Iliadis2024-1}.