Consider an o-minimal expansion of the real field. We deal
with the real spectrums of the ring $C_{\mathrm{df}}^r$ of
definable $C^r$ functions on an affine definable $C^r$ manifold
$M$ in the present paper. Here $r$ denotes a nonnegative integer.
We show that the natural map $\operatorname{Sper}(C_{\mathrm{df}}^r(M))
\rightarrow \operatorname{Spec}(C_{\mathrm{df}}^r(M))$ is a
homeomorphism when the o-minimal structure is polynomially
bounded. If the o-minimal structure is not polynomially bounded,
it is not known whether the natural map $\operatorname{Sper}(C_{\mathrm{df}}^r(M))
\rightarrow \operatorname{Spec}(C_{\mathrm{df}}^r(M))$ is a
homeomorphism or not. However, the natural map $\operatorname{Sper}(C_{\mathrm{df}}^0(M))
\rightarrow \operatorname{Spec}(C_{\mathrm{df}}^0(M))$ is bijective
even in this case.