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# Fort Topology Watch

1. Hi Everyone

The Fort topology is defined by:

is finite, or

If is non empty and I need to show that and are equivalent.

The homeomorphism I constructed is as follows:

This is useful (I hope!) because now but how does one show that f is continuous?

Usually we use this theorem: "f is continuous iff

So if we assume then either:

Case 1: or
Case 2: is finite with .

Case 1 I think I've got sorted, but what about case 2? If the function simplified to then it would be easy: which is finite. But how do you show is finite in general?

Thanks
2. Note that your f is a bijection. So it shouldn't be hard to prove stuff about the size of f*(V)...
3. (Original post by DFranklin)
Note that your f is a bijection. So it shouldn't be hard to prove stuff about the size of f*(V)...
Is this kind of reasoning ok:

The cardinality of = the cardinality of so this implies that the cardinality of = the cardinality of ? And hence finite
4. No, you can't do that. Because V and f*(V) might both be infinite, and this doesn't let you deduce anything abou the relative sizes of V^c and (f*V)^c.

On the other hand, you can confidently say that V^c and f*(V^c) have the same size...
5. (Original post by DFranklin)
No, you can't do that. Because V and f*(V) might both be infinite, and this doesn't let you deduce anything abou the relative sizes of V^c and (f*V)^c.

On the other hand, you can confidently say that V^c and f*(V^c) have the same size...
Oooh thank you so much

And f*(V^c) = (f*(V))^c? ...Surely... ? And hence the pre image is finite?
6. Yes.

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Updated: November 28, 2010
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