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The topologies are different. See Dugundji's Topology, Appendix I, 4.3: if $S$ has cardinality $2^{\aleph_0}$ (or larger) then $\mathcal{C}$ is not locally convex. The box topology is generated by products of intervals and hence locally convex.
How should I read "2 implies 1"? "If for some $Y$ the space $C(X,Y)$ does not have an isolated point, then $C(X,X)$ does not have one"? Or "If for all $Y$ $\ldots$"?
That is not quite the gist. It states that the assumption on sets of cardinality $\lambda$ is not an assumption but a fact: since the $E^\nu_\xi$ are pairwise disjoint $\xi\mapsto \min(E^\nu_\xi\cap T)$ is an injection into $T$, where $\xi$ runs through the indices corresponding to a nonempty intersection. And second: Hajnal indicates how to prove this, by induction.
