Compactifications and property of Kelley in generalized inverse limits
Keywords:
Generalized inverse limits, compactifications of a ray, Kelley continuumAbstract
To the memory of Phil Zenor: A fine mathematician who enjoyed a full life and who brightened the lives of those around him.
In 1969, R. Bennett gave conditions on a continuous function $f:[0,1] \rightarrow[0,1]$ so the inverse limit $X$ with fixed bonding map $f$ is the compactification of a ray. Mahavier in 2004 relaxed the continuity assumption on $f$ to upper semicontinuous and continuum-valued to show $X$ is a compactification of a connected set $R$. We generalize Mahavier's result and give necessary and sufficient conditions for $R$ to be a ray. We also give conditions so $X$ has the property of Kelley if $X \backslash R$ does. These results partially answer questions posed by W. T. Ingram in 2012.
References
R. Bennett, On inverse limit sequences, M.S. Thesis, The University of Tennessee (1962).
G. W. Henderson, The pseudo-arc as an inverse limit with one bonding map, Duke Math. J. 31, no. 3 (1964), 421-425.
W. T. Ingram, Inverse limits of families of set-valued functions, Bol. Soc. Mat. Mex. 21 (2015), 53-70.
W. T. Ingram, Families of Inverse Limits on $[0,1]$, Topology Proc. 27, No. 1 (2003), 189-201.
W. T. Ingram, Inverse limits with upper semi-continuous bonding functions: problems and some partial solutions., Topology Proc. 36 (2010), 353-373.
W. T. Ingram, An Introduction to inverse limits with set-valued functions, Springer Briefs in Mathematics (2012).
W. T. Ingram, W. S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston Journal of Math. 32, No. 1 (2006), 119-130.
W. T. Ingram, W. S. Mahavier, Inverse Limits: From Continua to Chaos, Developments in Mathematics, 25, Springer New York (2012).
W. T. Ingram, Concerning Chainability of Inverse Limits on $[0,1]$ with Set-Valued Functions, Topology Proc. 42 (2013), 327-340.
C. Islas, A 2-equivalent Kelley Continuum, Glas. Mat. Ser. III 46(66), no. 1 (2011), 249-268.
W. Mahavier, Inverse limits with subsets of $[0,1] \times[0,1]$, Topology Appl, 141, no. 3 (2004), 225-231.
S. B. Nadler, Jr., Hyperspaces of Set: A text with research questions, Aportaciones Matemáticas, Serie de Textos, Nivel Avanzado, 33, S. M. M. (2006).
S. B. Nadler, Jr., Continuum Theory: An Introduction, Monographs and Textbooks in Pure and Applied Math., 158, Marcel Decker, New York, Basel, Hong Kong, 1992.
