Ballester, Coloma; Caselles, Vicent The \(M\)-components of level sets of continuous functions in WBV. (English) Zbl 0991.54013 Publ. Mat., Barc. 45, No. 2, 477-527 (2001). Author’s abstract: “It is proved that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of \(M\)-connected components of its level sets, coincides when the function is a continuous function in WVB. Both function spaces are frequently used as models for images. Thus, if the domain \(\text{cl}(\Omega)\) of the image is a Jordan domain, a rectangle, for instance, and the image \(u \in C(\text{cl}(\Omega)) \cap \text{WBV}(\Omega)\) (being constant near \(\delta \Omega \)), we prove that for almost all levels \( \lambda \) of \(u\), the classical connected components of positive measure of \([u \geq \lambda ]\) coincide with the \(M\)-components of \([u \geq \lambda ]\). Thus the notion of \(M\)-component can be seen as a relaxation of the classical notion of connected component when going from \(C(\text{cl}(\Omega))\) to \(\text{WBV} (\Omega)\)”. Reviewer: Zbigniew Grande (Bydgoszcz) Cited in 2 Documents MSC: 54C30 Real-valued functions in general topology 26A45 Functions of bounded variation, generalizations 28A99 Classical measure theory Keywords:level sets; connected components; Morse theory; functions of bounded variation; sets of finite perimeter PDFBibTeX XMLCite \textit{C. Ballester} and \textit{V. Caselles}, Publ. Mat., Barc. 45, No. 2, 477--527 (2001; Zbl 0991.54013) Full Text: DOI EuDML