# Definition: Transition Map

Let $$M$$ be a manifold and let $$U_{\alpha},U_{\beta}\subseteq M$$ be its open subsets. Furthermore, let $$V_\alpha,V_\beta\subseteq\mathbb R^{n}$$ be open subsets of the $$n$$-dimensional metric space or real numbers $$\mathbb {R} ^{n}$$.

Consider two topological charts $$\varphi_{\alpha}\colon U_{\alpha}\rightarrow V_{\alpha}$$ and $$\varphi_{\beta}\colon U_{\beta}\rightarrow V_{\beta}$$. Then we call a composition of the homeomorphisms. $\tau_{\alpha,\beta}:=\varphi_{\beta}\circ \varphi_{\alpha}^{-1}=V_{\alpha}\cap \varphi_{\alpha}(U_{\alpha}\cap U_{\beta})\longrightarrow V_{\beta}\cap \varphi_{\beta}(U_{\alpha}\cap U_{\beta})$

the transition map of the topological charts.

Note that since $$\varphi _{\alpha }$$ and $$\varphi _{\beta }$$ are both homeomorphisms, the transition map $$\tau _{\alpha ,\beta }$$ is also a homeomorphism.

(picture by Stomatapoll, wikipedia)

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### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of OsnabrÃ¼ck