Conditions for continuity of certain open monotone functions

Author:
Melvin R. Hagan

Journal:
Proc. Amer. Math. Soc. **30** (1971), 175-178

MSC:
Primary 54.60

DOI:
https://doi.org/10.1090/S0002-9939-1971-0279779-X

MathSciNet review:
0279779

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Abstract: In this paper continuity of certain open monotone functions is obtained by assuming for the domain and/or range various combinations of the properties of a metric continuum, regular metric continuum, semilocal connectedness, and hereditary local connectedness. An open monotone connected function from a hereditarily locally connected separable metric continuum onto a separable metric continuum is continuous. If the domain is a regular separable metric continuum, an upper semicontinuous decomposition and resulting monotone-light factorization yield continuity of an open monotone function with closed point inverses.

**[1]**M. R. Hagan,*A note on connected and peripherally continuous functions*. Proc. Amer. Math. Soc.**26**(1970), 219-223. MR**0263042 (41:7647)****[2]**P. E. Long,*Properties of certain non-continuous transformations*, Duke Math. J.**28**(1961), 639-645. MR**24**#A2945. MR**0133111 (24:A2945)****[3]**-,*Connected mappings*, Duke Math. J.**35**(1968), 677-682. MR**38**#2745. MR**0234428 (38:2745)****[4]**G. T. Whyburn,*Analytic topology*, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R. I., 1963. MR**32**#425. MR**0182943 (32:425)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0279779-X

Keywords:
Open monotone function,
connected function,
connectivity function,
peripherally continuous function,
regular continuum,
semilocally connected,
hereditarily locally connected,
monotone-light factorization,
upper semicontinuous decomposition

Article copyright:
© Copyright 1971
American Mathematical Society