Given a metric space \(X\) the authors are dealing with function spaces \(C(X)\) of real-valued continuous functions endowed with a uniform convergence \(\mathcal{T}_\mathcal{B}\) with respect to some given bornology \(\mathcal{B}\) on \(X\), thus generalizing pointwise convergence, uniform convergence on compacta and uniform convergence. In total four topologies on \(C(X)\) are considered, besides \(\mathcal{T}_\mathcal{B}\), also the Whitney topology \(\mathcal{T}^w_\mathcal{B}\) introduced by H. Whitney [Ann. Math. (2) 37, 645–680 (1936; Zbl 0015.32001)], the strong Whitney convergence \(\mathcal{T}^{sw}_\mathcal{B}\) introduced by A. Caserta [Filomat 26, No. 1, 81–91 (2012; Zbl 1289.54022)] and the strong uniform convergence \(\mathcal{T}^{s}_\mathcal{B}\) introduced by G. Beer and S. Levi [J. Math. Anal. Appl. 350, No. 2, 568–589 (2009; Zbl 1161.54003)]. The concept of a shield introduced by G. Beer et al. [Mediterr. J. Math. 10, No. 1, 529–560 (2013; Zbl 1275.54003)] plays an important role in the present study of \(\mathcal{T}^{sw}_\mathcal{B}\) and \(\mathcal{T}^{s}_\mathcal{B}\) thus yielding new characterizations for the notion of a shield.
As a starting point the authors define particular linear subspaces of \(C(X),\) which appears to be an important step in obtaining characterizations of the clopen linear subspaces of \(C(X)\) endowed with any of the four topologies considered.
Finally it is shown that \(C(X)\) endowed with any of the four topologies need not be connected and hence need not be a topological vector space. Necessary and sufficient conditions are given for the function spaces under study to be connected and locally convex.