## Countable-dimensional spaces: A survey.(English)Zbl 0541.54042

This is a monograph for countable-dimensional spaces with many well organized questions. A space means a completely regular space and the dimension is the covering dimension defined by means of finite cozero covers originated by E. Čech. A space X is said to be countable- dimensional (c.d.) if X is the union of a sequence $$X_ 1,X_ 2,..$$. of subspaces with dim $$X_ i<\infty$$ for each i. According to the reviewer [Proc. Japan Acad. 41, 771-772 (1965; Zbl 0141.400)], a metric space X is c.d. if and only if X is the union of a sequence $$X_ 1,X_ 2,..$$. of subspaces with $$\dim X_ i\leq 0$$ and with $$X_ i\subset X_{i+1}$$ for each i. According to V. V. Fedorchuk [Izv. Akad. Nauk SSSR, Ser. Mat. 42, 1163-1178 (1978; Zbl 0403.54025); Math. USSR, Izv. 13, 445- 460 (1979; Zbl 0423.54024)], under the assumption of the Continuum Hypothesis, there exists a perfectly normal compact c.d. space X with $$\dim X=2$$ which cannot be the union of a sequence of 0-dimensional subspaces. By this observation of a narrower subclass of c.d. spaces is defined as follows: A space is said to be zero-countable-dimensional (0- c.d.) if it is the union of a sequence of 0-dimensional subspaces. (1) Define, without any additional set-theoretic assumptions, a hereditarily normal c.d. space which is not 0-c.d. (2) Give an example of a hereditarily normal space X with dim X$$=1$$ which is not 0-c.d. (3) Give an example of a hereditarily normal 0-c.d. space which cannot be the union of an increasing sequence of 0-dimensional subspaces. A space X is said to be strongly countable-dimensional (s.c.d.) if it is the union of a sequence of closed subspaces $$X_ 1,X_ 2,..$$. with dim $$X_ i<\infty$$ for each i. R. Pol [Colloq. Math. 44, 65-76 (1981; Zbl 0479.54008)] gave an example of a separable metric space X with dim X$$=1$$ which can be the union of a transfinite sequence $$X_{\alpha}$$, $$\alpha<\omega_ 1$$, of closed subspaces with $$\dim X_{\alpha}\leq 0$$ for each $$\alpha<\omega_ 1$$. (4) Is every homogeneous compact metric c.d. space necessarily finite-dimensional? (5) Does every homogeneous compact metric space that contains the n-cube $$I^ n$$ for $$n=1,2,..$$. necessarily contain the Hilbert cube? Under the assumption of the Continuum Hypothesis there exists a hereditarily normal compact space X with $$\dim X=0$$ which contains a subspace which is not c.d. (6) Define, without any set-theoretic assumptions, a hereditarily normal c.d. (0- c.d.) space that contains a subspace which is not c.d. (0-c.d.). (7) Let X be a hereditarily normal space such that every open subspace is c.d. (0-c.d., s.c.d.). Then is every subspace c.d. (0-c.d., s.c.d.)? These questions are some of 27 questions concerning the following items: Subspace theorems, addition and sum theorems, product theorems, compactification and completion theorems, universal space theorems, mapping theorems, relations to other classes of infinite-dimensional spaces. Each problem is offered with reasonable and kind illustration of its background.
Reviewer: K.Nagami

### MSC:

 54F45 Dimension theory in general topology 54-02 Research exposition (monographs, survey articles) pertaining to general topology

### Citations:

Zbl 0141.400; Zbl 0403.54025; Zbl 0423.54024; Zbl 0479.54008