## Spaces having a weakly-infinite-dimensional compactification.(English)Zbl 0587.54055

The author introduces the property, small-weakly-infinite-dimensional (small-w.i.d.). He proves that for a separable metric space X, the following are equivalent, (1) X is small-w.i.d., (2) X has a small-w.i.d. completion, (3) X has a w.i.d. metric compactification.
He shows that every complete, separable metric, totally disconnected space is small-w.i.d. Since it is known that there exist complete, separable metric, totally disconnected spaces that are infinite dimensional but not countable dimensional, then the preceding shows that there exists a metric compactum which is neither countable dimensional nor strongly infinite dimensional. This is of course the result due to R. Pol [Proc. Am. Math. Soc. 82, 634-636 (1981; Zbl 0469.54014)].
Reviewer: L.Rubin

### MSC:

 54F45 Dimension theory in general topology 55M10 Dimension theory in algebraic topology

Zbl 0469.54014
Full Text:

### References:

 [1] Engelking, R., Dimension theory, (1978), PWN Warszawa [2] Engelking, R.; Pol, E., Countable dimensional spaces. A survey, Dissertationes mathematicae, CCXVI, 1-45, (1983) · Zbl 0496.54032 [3] Hurewicz, W., Über unendlich-dimensionale punktmengen, Proc. akad. Amsterdam, 31, 916-922, (1928) · JFM 54.0620.05 [4] Lelek, A., On the dimension of remainders in compact extensions, Soviet math. dokl., 6, 136-140, (1965) · Zbl 0134.18802 [5] Misra, A.K., Some regular wallman βX, Indag. math., 35, 237-242, (1973) · Zbl 0258.54022 [6] Pol, R., A weakly infinite-dimensional compactum which is not countable dimensional, Proc. amer. math. soc., 82, 634-636, (1981) · Zbl 0469.54014 [7] Schurle, A.W., Compactification of strongly countable dimensional spaces, Trans. amer. math. soc., 136, 25-32, (1969) · Zbl 0175.19902
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