Weakly infinite-dimensional space
A topological space $X$ such that for any infinite system of pairs of closed subsets $(A_i,B_i)$ of it, $$ A_i \cap B_i = \emptyset,\ \ i=1,2,\ldots $$ there are partitions $C_i$ (between $A_i$ and $B_i$) such that $\cap C_i = \emptyset$. An infinite-dimensional space which is not weakly infinite dimensional is called strongly infinite dimensional. Weakly infinite-dimensional spaces are also called $A$-weakly infinite dimensional. If in the above definition it is further required that some finite subfamily of the $C_i$ have empty intersection, one obtains the concept of an $S$-weakly infinite-dimensional space.
|||P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)|
In addition to the above, $A$-weakly stands for Aleksandrov weakly, and $S$-weakly for Smirnov weakly. There is also the obsolete notion of Hurewicz-weakly infinite-dimensional space. Cf. the survey [a1].
To avoid ambiguity in the phrase "infinite-dimensional space" , the space $X$ could be required to be metrizable, cf. [a2].
See also Infinite-dimensional space.
|[a1]||P.S. Aleksandrov, "Some results in the theory of topological spaces, obtained within the last twenty-five years" Russian Math. Surveys , 15 : 2 (1960) pp. 23–83 Uspekhi Mat. Nauk , 15 : 2 (1960) pp. 25–95|
|[a2]||J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1989) pp. 40|
|[a3]||R. Engelking, E. Pol, "Countable-dimensional spaces: a survey" Diss. Math. , 216 (1983) pp. 5–41|
Weakly infinite-dimensional space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weakly_infinite-dimensional_space&oldid=42009