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I have seen in a few abstracts, as this for instance: A survey of topological games the remark that the subject Topological games has applications in other fields of mathematics. I am familiar with topological games in selection principles. But still. I can't see the advantage in describing a situation by means of a topological game rather them by simply describe the topological properties of the space.

Does anyone have a relatively simple answer for that? What are the advantages of describing a topological space by a game rather then by it's topological properties.

By, relatively simple, I mean, an answer in a level of a graduate student which is familiar with concepts of topology and set theory but is not, yet, in a research level/

Thank you!

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up vote 3 down vote accepted

Generally speaking, one can say that topological games are a very powerful tool in the area of descriptive set theory.

Typically, whenever you want to prove a "dichotomy result", i.e. a statement of the form "either something happens, or a strong form of the converse happens", it is a good idea to try to prove it by using a game. The "something" will happen if one of the players has a winning strategy, and the strong converse will happen if the other player has a winning strategy. So, if the game is determined, the dichotomy holds true.

For example, one can prove with the help of a suitable game that if $A$ is a Borel subset of $\mathbb R$, then either $A$ is countable, or $A$ is uncountable in a strong sense, namely it contains a perfect set (a nonempty closed set without isolated points). This is the so-called "perfect set theorem" for Borel sets.

If you are interested in these kinds of ideas, I suggest that you have a look at Kechris' book Classical Descriptive Set theory. In particular, sections 8.C, 8.H and Chapter 21.

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ok I will. Thank you! – topsi Feb 27 '14 at 11:01
You're welcome. I hope you'll enjoy the reading. – Etienne Feb 27 '14 at 11:49

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