Correction to `` Some aspects of real-analytic manifolds and differentiable manifolds''
Kōji Shiga · Journal of the Mathematical Society of Japan · 1965
In [2; Appendix, p. 139] we stated an extension theorem of $C^{s}$ -mappings which was necessary to the proof of Classification Theorem of $C^{S}-fibre$ bundles $(1 \leqq s\leqq\omega)$ .However, the proof given there was incorrect and our reference to [3] was not pertinent to this theorem.Now we give a proof of the fol- lowing extension theorem which corrects the theorem stated in [2, p. $ 139\rfloor$ .THEOREM.Let $M$ and $N$ be C'-manifolds, and let $L$ be a closed $C^{\omega}$ -submanifold of M. Suppose that we have a $C^{\omega}$ -mapping $\varphi$ of $L$ into $N$ such that $\varphi$ can be extended to a $C^{0}$ -mapping $f$ of $M$ into N.Then, for any positive family 8, there exists a $C^{\omega}$ -mapping $\psi$ from $M$ into $N$ having the following properties:(i) $\psi$ gives an $\mathcal{E}$ -approximation to $f$ in order $0$ .(ii) $\psi(p)|L=\varphi(p)$ .Here we formulate only real-analytic case, because differentiable case is trivial [cf.2, p. 140].If this theorem is established, then Theorem $C[2,$ $p$ . 138]remains valid.