By Bacco M., Mocellin V.

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Functions defined on preimages of Euclidean spheres . . . . . . Functions on spaces including R . . . . . . . . . . . . . . . . . Functions on compact subsets of Euclidean spaces . . . . . . . Peano maps and space-filling curves . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and remarks . . . . . . . . . . . . . . . . . . . .

Baire’s theorem provides a kind of space in which this cannot happen. 4. If X is a Baire space space then X is of second category in itself. In particular, any completely metrizable topological space is of second category in itself. Proof. Assume, by way of contradiction, that X is meager. There there exist ∞ rare subsets Rn such that X = n=1 Rn . Then, setting Fn := Rn (n ≥ 1), ∞ it follows that X = n=1 Fn , where each Fn is a closed subset satisfying Fn0 = ∅. Let An := Fnc . Then each An is open and dense and, consequently, ∞ ∅ = n=1 An is dense, which is absurd.

N, as required. 5 Surjections, Darboux functions and related properties The class of Darboux functions is, probably, the one that has produced the most lineability results within the framework of real analysis. 1). This section does not pretend to cover all results within 40 Lineability: The search for linearity in Mathematics this vast class, since there are (at the moment) ongoing research within it. However, we refer to [167, 323] for some recent advances inside this class. We would like to emphasize that this section simply pretends to give a general overview of some lineability results concerning the mentioned class.