I am very early into my first look at topology (specifically, I am jumping to topology on smooth manifolds through the Baez/Muniain book - Gauge Fields, Knots and Gravity) and I have a few questions. Suppose we have overlapping sets on a manifold A and B like this(adsbygoogle = window.adsbygoogle || []).push({});

where A and B are both open. It can be shown that the union of A and B is also open, which sort of follows intuitively when looking at the boundaries. This means that, when you approach the 'boundary' from a point x contained in A union B, you can obtain a point in between x and the 'boundary' that is still contained in the union. But by applying the idea that a complement of an open set is a closed set, we see that when we approach from a point only within B, we can reach an edge where there is no other point inside B that is between the boundary and the original point. Same argument for A. The collection of these points would be the boundary. Check me if this reasoning isn't correct.

My question is in regards to the use of the smooth transition function [tex]\phi_{A}\phi_{B}^{-1}:R^{n}\rightarrow R^{n}[/tex]. I am having trouble really nailing down what the transition function does. It seemed to me at first that

[tex]\phi_{A}\phi^{-1}_{B}=\phi_{B}\phi^{-1}_{A}[/tex]

But after thinking about it some more, that would lead to

[tex]\phi_{A}=\phi_{B}[/tex]

Which doesn't make any sense, if they were equal to each other then there would be no need for a transition function. So this implies to me that for each patch on the manifold there is an entirely new [tex]R^{n}[/tex] associated with it. So when there is a union between two patches A and B, it is possible/probable that the various regions within that union would be mapped to different places in each a different [tex]R^{n}[/tex]. What I mean is that there is a sort of indexing of the real spaces going on, for A there is [tex]R^{n}_{A}[/tex] and for B there is [tex]R^{n}_{B}[/tex], which is another nth dimensional space that is not associated with the others except by the transition functions.

Is this all correct reasoning, if so I am set and question answered!

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# Closed/Open Transition Function Question

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