  Based on the observation that the skyrmion in Skyrme theory can be viewed as
a dressed monopole, we show that the skyrmions have two independent topology,
the baryon topology $\pi_3(S^3)$ and the monopole topology $\pi_2(S^2)$. With
this we propose to classify the skyrmions by two topological numbers $(m,n)$,
the monopole number $m$ and the shell (radial) number $n$. In this scheme the
popular (non spherically symmetric) skyrmions are classified as the $(m,1)$
skyrmions but the spherically symmetric skyrmions are classified as the $(1,n)$
skyrmions, and the baryon number $B$ is given by $B=mn$. Moreover, we show that
the vacuum of the Skyrme theory has the structure of the vacuum of the
Sine-Gordon theory and QCD combined together, which can also be classified by
two topological numbers $(p,q)$. This puts the Skyrme theory in a totally new
perspective.
