Some special decompositions of 𝐸³

Bass, Charles D.

doi:10.1090/S0002-9947-1976-0405425-7

Some special decompositions of $E^{3}$
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by Charles D. Bass PDF

Trans. Amer. Math. Soc. 216 (1976), 115-130 Request permission

Abstract:

A great deal of attention has been given to the question: which upper semicontinuous decompositions of ${E^3}$ into pointlike continua give ${E^3}$. It has recently been determined that some decompositions of ${E^3}$ into points and straight line segments give decomposition spaces which are topologically distinct from ${E^3}$. In this paper we apply a new condition to the set of nondegenerate elements of a decomposition which enables one to conclude that the resulting decomposition space is homeomorphic to ${E^3}$.

References

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R. H. Bing, Upper semicontinuous decompositions of $E^3$, Ann. of Math. (2) 65 (1957), 363–374. MR 92960, DOI 10.2307/1969968

Decompositions of

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E. Dyer and M.-E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1958), 103–118. MR 92959, DOI 10.4064/fm-45-1-103-118
W. T. Eaton, Applications of a mismatch theorem to decomposition spaces, Fund. Math. 89 (1975), no. 3, 199–224. MR 391095, DOI 10.4064/fm-89-3-199-224
Louis F. McAuley, Some upper semi-continuous decompositions of $E^{3}$ into $E^{3}$, Ann. of Math. (2) 73 (1961), 437–457. MR 126258, DOI 10.2307/1970312

Upper semicontinuous decompositions of

into

and generalizations to metric spaces

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