Metrizability of general ANR

Tsuda, Kōichi

doi:10.1090/S0002-9939-1986-0818475-5

Metrizability of general ANR
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by Kōichi Tsuda PDF

Proc. Amer. Math. Soc. 96 (1986), 375-378 Request permission

Abstract:

We show that every nonmetrizable ${\text {ANR(}}\mathcal {P}{\text {)}}$ contains a copy of a Tychonoff cube of uncountable weight. Hence, every finite dimensional ${\text {ANR(}}\mathcal {P}{\text {)}}$ is metrizable, and every ${\text {ANR(}}\mathcal {P}{\text {)}}$, each point of which is a ${G_\delta }$-set, is metrizable, where $\mathcal {P}$ denotes the class of all paracompact $p$-spaces.

References

Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
R. Engelking, On functions defined on Cartesian products, Fund. Math. 59 (1966), 221–231. MR 203697, DOI 10.4064/fm-59-2-221-231
Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
B. S. Klebanov, The metrization of $\varphi$-spaces, Dokl. Akad. Nauk SSSR 246 (1979), no. 4, 809–812 (Russian). MR 543537
Jun-iti Nagata, A note on $M$-space and topologically complete space, Proc. Japan Acad. 45 (1969), 541–543. MR 254803
T. Przymusiński, Collectionwise normality and absolute retracts, Fund. Math. 98 (1978), no. 1, 61–73. MR 528355, DOI 10.4064/fm-98-1-61-73
E. V. Ščepin, A finite-dimensional bicompact absolute neighborhood retract is metrizable, Dokl. Akad. Nauk SSSR 233 (1977), no. 3, 304–307 (Russian). MR 0487966
Eugène Ščepin, Sur les applications continues des cubes de Tihonov, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 4, A257–A260 (French, with English summary). MR 524787
K\B{o}ichi Tsuda, A note on Ščepin’s theorem, Proc. Amer. Math. Soc. 91 (1984), no. 1, 167–170. MR 735586, DOI 10.1090/S0002-9939-1984-0735586-1
Yukinobu Yajima, The normality of $\Sigma$-products and the perfect $\kappa$-normality of Cartesian products, J. Math. Soc. Japan 36 (1984), no. 4, 689–699. MR 759424, DOI 10.2969/jmsj/03640689