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

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]**Melvin R. Hagan,*A note on connected and peripherally continuous functions*, Proc. Amer. Math. Soc.**26**(1970), 219–223. MR**0263042**, 10.1090/S0002-9939-1970-0263042-6**[2]**Paul E. Long,*Properties of certain non-continuous transformations*, Duke Math. J.**28**(1961), 639–645. MR**0133111****[3]**Paul E. Long,*Connected mappings*, Duke Math. J.**35**(1968), 677–682. MR**0234428****[4]**Gordon Thomas Whyburn,*Analytic topology*, American Mathematical Society Colloquium Publications, Vol. XXVIII, American Mathematical Society, Providence, R.I., 1963. MR**0182943**

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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