Concerning 𝑛-mutual aposyndesis in products of continua

Rogers, Leland E.

doi:10.1090/S0002-9947-1971-0293599-6

Concerning $n$-mutual aposyndesis in products of continua
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Trans. Amer. Math. Soc. 162 (1971), 239-251 Request permission

Abstract:

This paper is concerned with Cartesian products of regular Hausdorff continua and certain conditions on the factors that make the product n-mutually aposyndetic (given n distinct points, there are n disjoint subcontinua, each containing one of the points in its interior). It is proved that the product of any three regular Hausdorff continua is n-mutually aposyndetic for each $n \geqq 2$. Next, certain conditions on factors of products of two continua are shown to be sufficient for the product to be n-mutually aposyndetic. In connection with this, the concepts of n-semiaposyndesis and aposyndetic-terminal points are introduced. Finally, it is proved that the product of a simple closed curve (or any other “super n-mutually aposyndetic” continuum) with every compact Hausdorff continuum is n-mutually aposyndetic for each $n \geqq 2$.

References

R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43–51. MR 43451

Homogeneous continua

H. S. Davis, D. P. Stadtlander, and P. M. Swingle, Properties of the set functions $T^{n}$, Portugal. Math. 21 (1962), 113–133. MR 142108
R. W. FitzGerald, The cartesian product of non-degenerate compact continua is $n$-point aposyndetic, Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968, pp. 324–326. MR 0236861
J. B. Fugate, Decomposable chainable continua, Trans. Amer. Math. Soc. 123 (1966), 460–468. MR 196720, DOI 10.1090/S0002-9947-1966-0196720-1
C. L. Hagopian, Mutual aposyndesis, Proc. Amer. Math. Soc. 23 (1969), 615–622. MR 247612, DOI 10.1090/S0002-9939-1969-0247612-9

Concerning the cyclic connectivity of plane continua

Charles L. Hagopian, Concerning arcwise connectedness and the existence of simple closed curves in plane continua, Trans. Amer. Math. Soc. 147 (1970), 389–402. MR 254823, DOI 10.1090/S0002-9947-1970-0254823-8
Charles L. Hagopian, Arcwise connectedness of semiaposyndetic plane continua, Trans. Amer. Math. Soc. 158 (1971), 161–165. MR 284981, DOI 10.1090/S0002-9947-1971-0284981-1
Charles L. Hagopian, An arc theorem for plane continua, Illinois J. Math. 17 (1973), 82–89. MR 314010
F. Burton Jones, Aposyndetic continua and certain boundary problems, Amer. J. Math. 63 (1941), 545–553. MR 4771, DOI 10.2307/2371367
F. Burton Jones, Concerning non-aposyndetic continua, Amer. J. Math. 70 (1948), 403–413. MR 25161, DOI 10.2307/2372339
F. B. Jones, Concerning the boundary of a complementary domain of a continuous curve, Bull. Amer. Math. Soc. 45 (1939), no. 6, 428–435. MR 1563999, DOI 10.1090/S0002-9904-1939-07000-6
R. L. Moore, Foundations of point set theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR 0150722
Leland E. Rogers, Mutually aposyndetic products of chainable continua, Pacific J. Math. 37 (1971), 805–812. MR 365522
R. H. Sorgenfrey, Concerning triodic continua, Amer. J. Math. 66 (1944), 439–460. MR 10968, DOI 10.2307/2371908
E. S. Thomas Jr., Monotone decompositions of irreducible continua, Rozprawy Mat. 50 (1966), 74. MR 196721
Eldon Jon Vought and F. Burton Jones, Stronger forms of aposyndetic continua, Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968, pp. 170–173. MR 0239568
G. T. Whyburn, Semi-locally connected sets, Amer. J. Math. 61 (1939), 733–749. MR 182, DOI 10.2307/2371330

Analytic topology

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