Exercise 7, § 2

a) Let R be a set. If S is a subset of R, we denote by $j_S$ the mapping of $\mathbf{I}^S$ into $\mathbf{I}^R$ which maps a function $f: S\rightarrow \mathbf{I}$ onto the unique function from R to $\mathbf{I}$ whose restriction to S is equal to $f$ and which is zero at every point of R-S.

Prove that $j_S$ is a homeomorphism of $\mathbf{I}^S$ onto the subspace of $\mathbf{I}^R$ formed by the functions which are zero outside S.

b) Let X be the set of mappings from R into $\mathbf{I}$ which are zero outside a countable subset of R. We denote by Y the inductive limit of the family of spaces $(\mathbf{I}^S)$, where S runs through the set of countable subsets of R, and we equip it with the inductive limit topology of the product topologies.

The canonical mappings of $\mathbf{I}^S$ into $\mathbf{I}^{\mathbf{R}}$ induce a continuous bijection $j$ from Y onto X. This bijection is not a homeomorphism if R is not countable.

c) Prove that, for every path $c:\mathbf{I}\rightarrow X$, there exists a countable subset S of R such that $c(t)(s) = 0$ for every $t\in \mathbf{I}$ and every $s\in R-S$. d) Deduce that the mapping $j^{-1}$ is continuous by arcs.