What do we mean by orientability of a topological manifold? How do we orient two dimensional Euclidean space and why is Moebius band non-orientable? And it would be a great favour to me if you can tell me a good book that explains this idea. I read "Algebraic topology: An Introduction" by W. S. Massey but have a confusion.
To answer your question about Euclidean two-space first: If $V$ is a vector space and $e_i$ and $f_i$ are two bases then they are said to have the same orientation if the determinant of the matrix expressing one basis in terms of the other is positive that is, $>0$.
The relation you define in such a way on the set of all bases of $V$ is an equivalence relation and consists of two equivalence classes.
Hence you may define the orientation of a vector space as a choice of an equivalence class of bases.
See e.g. Boothby on page 207/208.
Why is the Moebius band not orientable? In the case of topological manifold one can choose a normal vector. Say you start at a point $a$ on the Moebius band and pick a vector pointing upwards. If you slide this vector along the middle circle of the band one time around it will point the opposite way. A manifold where you can continuously move around a point from "one orientation" into another different one is non-orientable.
See e.g. this post which explains it nicely.
Another way of defining it is given in Hatcher starting on page 233.
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