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This talk gives a topological version of Oseledec's multiplicative ergodic theorem that is the fundamental theorem in the smooth ergodic theory. Oseledec's theorem is stated in the measure-theoretic setting, and Oseledec decompositions of tangent bundles are measure-theoretic splittings. We will construct continuous splittings of tangent bundles, in which iterations of the derivatives behave like 1-dimensional linear maps on each subbundle with respect to some time series. The base spaces of our setting are subsets like minimal sets. In the case where the base space is chosen as a periodic orbit, our splitting is consistent with Jordan normal form.