$\mathbf{c}^{+}$-Lineability of the Class of Darboux Maps with the Strong Cantor Intermediate Value Property Which Are Not Connectivity
Keywords:
connectivity maps, Darboux property, lineability, SCIVPAbstract
We prove, under an additional set-theoretic assumption, specifically continuum is a regular cardinal, that there exists a subspace of the vector space $\mathbb{R}^{\mathbb{R}}$ of dimension $c^{+}$whose non-zero elements are the functions that are everywhere surjective (ES), have strong Cantor intermediate value property (SCIVP), and are not connectivity (Conn). Since every map in ES is Darboux (D), this means that the class SCIVP $\cap \mathrm{D} \backslash$ Conn is $c^{+}$-lineable under our set-theoretic assumption.
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