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As we know, there is some relationship, such as precedence relation, among skills. Each precedence relation induces a competence structure. Thus, we study competence-based skill functions, which rely on competence structures and go from somethings observable to somethings invisible. Conversely, competence-based problem functions go from somethings invisible to somethings observable. In fact, these two dual types of functions based on competence structures are symmetry. Remarkably, there are two kinds of special competence-based skill functions: one is disjunctive, while the other is conjunctive. The former delineates knowledge spaces, which are symmetrical to simple closure spaces delineated by the latter. Based on these facts, we shows some theoretical results on competence-based skill functions, then design the corresponding algorithms for delineating knowledge structures. Sometimes for competence-based skill functions, some skills are maybe reducible. Thus, we discuss what kind of skills are reducible and obtain sufficient and some necessary conditions for skills being reducible for competence-based skill functions. Based on this, we design algorithms to reduce reducible skills and get minimal sets of skills. By comparison, for competence-based skill functions, we can find minimal sets of skills with the smallest cardinality whenever sets of skills are finite. For each algorithm, we take a corresponding example to illustrate the detailed procedure.

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Let C(I) be the set of all continuous self-maps from I = [0, 1] with the topology of uniformly convergence. A map f ∈ C(I) is called a transitive map if for every pair of non-empty open sets U, V in I, there exists a positive integer n such that U ∩ f −n (V ) = ∅. We note T (I) and T (I) to be the sets of all transitive maps and its closure in the space C(I). In this paper, we show that T (I) and T (I) are homeomorphic to the separable Hilbert space 2 .

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