Absolute end points of irreducible continua. (English) Zbl 0795.54047

Charatonik, Janusz J.

Math. Bohem. 118, No. 1, 19-28 (1993).

The concept of an absolute end point introduced for arc-like continua by Rosenholtz is extended to arbitrary irreducible continua. The point \(p\) of the continuum \(X\) is said to be an end point of \(X\) if and only if whenever two subcontinua of \(X\) contain \(p\) then one is a subset of the other. The point \(p\) of the continuum \(X\) is said to be an absolute end point if and only if \(X \backslash \{p\}\) is a composant of \(X\). The author observes that many of the Rosenholtz proofs carry over to the general setting. Relationships between end points, absolute end points, and points at which a given continuum is smooth are considered. The paper ends with the following theorem: Let a continuum \(X\) be irreducible between a point \(p\) and some other point of \(X\). If \(X\) is smooth at \(p\), then \(p\) is both an absolute end point and an end point of \(X\).

Reviewer: M.Smith (Auburn)

MSC:

54F15	Continua and generalizations
54F50	Topological spaces of dimension \(\leq 1\); curves, dendrites

Keywords:

bend set; pairwise smooth continuum; contractible continuum; curve; dendroid; fan; homotopy; pointwise smooth continuum; \(Q\)-point; \(R\)- continuum; selectible continuum; type \(N\); absolute end point