Subshift of Finite Type - Topology

Topology

The subshift of finite type has a natural topology, derived from the product topology on, where

V^\mathbb{Z}= \Pi_{n \in \mathbb{Z}} V = \{ x=(\ldots,x_{-1},x_0,x_1,\ldots) : x_k \in V \; \forall k \in \mathbb{Z} \}

and V is given the discrete topology.

A basis for the topology of the shift of finite type is the family of cylinder sets

C_t= \{x \in V^\mathbb{Z} : x_t = a_0, \ldots ,x_{t+s} = a_s \}

The cylinder sets are clopen sets. Every open set in the subshift of finite type is a countable union of cylinder sets. In particular, the shift T is a homeomorphism; that is, with respect to this topology, it is continuous with continuous inverse.

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