First, don't consider the topology, I can treat the $T^2$ and Klein bottle as giving a $S^1$ at evey point of $S^1$. And the Cartesian product has nothing to topology. So $T^2=S^1\times S^1=$ Klein bottle up to set.
Second ,if the $S^1\times S^1$ carry product topology , $(U,V)$ is open if $U,V$ open, then it is $T^2$ . But I feel that for $(U,V)\subset$Klein bottle , also $(U,V)$ is open if $U,V$ open . Then $T^2$ and Klein bottle have "same open set"(the words is not accurate, but I don't know how to describ it .). Obviously, it is wrong. Where is the mistake?
Third, if treat Klein bottle as $S^1\times S^1$ , then ,how to define its open sets making it is Klein bottle ?
I think you're thinking too much about set theory. You should only think about the topological properties of spaces when doing topology; it's evil to identify the underlying sets of topological spaces unless the bijection is a homeomorphism.
The Klein bottle and the Torus are both 2-manifolds, so obviously they look the same locally, and both can be given an $S^1$-bundle structure over $S^1$, so locally they are the same as fibre bundles. Note that a space is not "canonically" a bundle. A bundle is a space equipped with a projection map to a base so that it is locally trivial. A given space can be a bundle in different ways.
With regards to describing the open sets of the Klein bottle and the Torus, the most straightforward way to do it is by looking at the way they are typically constructed: as quotients of a square. The quotient construction tells you what the open sets look like.
The wikipedia article explains how you identify the edges of the square to get a Klein bottle. Notice the difference between the identification you use to get the torus: one of the edges is given a "twist" when it is glued to the other side.
If you want to describe the $S^1$-bundle structure of the Klein bottle, what you really ought to do it is describe the Cech cocycle of the bundle: (an equivalence class of) open coverings of the base with transition functions that satisfy the cocycle condition. Have a look in chapter 6 of Jeffrey Lee's text on Manifolds and Differential Geometry for a good introduction to fibre bundles and how you build them from cocycles.