  Fisher's phenomenological renormalization method is used to calculate the
mass gap and the correlation length of the $O(N)$ nonlinear $\sigma$ model on a
semi-compact space $S^{1}\times {\bf R}^{2}$. This shows that the ultraviolet
momentum cut-off does not conflict with the infrared cut-off along the $S^{1}$
direction. The mass gap on $S^{2}\times {\bf R}$ is also discussed.
