The shark teeth is a topological IFS-attractor (Q483446)

This paper answers a question raised in [\textit{T. Banakh} and \textit{M. Nowak}, Proc. Am. Math. Soc. 141, No. 3, 931--935 (2013; Zbl 1262.28001)]: is there an attractor for a topological iterated function system (TIFS) that is not an attractor (or homeomorphic to an attractor) of an iterated function system (IFS)? The answer is yes, the so-called shark teeth. A family of which are constructed and a particular shark teeth fractal is the example of such a fractal. The particular notion of a TIFS is given by the first author in [Zbl 1262.28001]. A TIFS is a finite collection of continuous functions, \(\{f_i\}\). A non-empty compact subset \(K\) is the attractor of a TIFS if for each open cover of \(K\) there exists an integer \(m\) such that \(f_{i_1} \circ \cdots \circ f_{i_m}(K)\) lies in some element of the open cover. This notion is different than other notions of a TIFS, for example the one in [\textit{N. A. Selecean}, Filomat 24, No. 4, 147--161 (2010; Zbl 1265.28025)] which is based on compact sets rather than open covers. The paper is compactly written and provides a clear answer to its question.