Published online by Cambridge University Press: 05 June 2014
We prove Tychonoff's theorem, that the topological product of compact topological spaces is compact. The key idea is that of a filter. This generalizes the notion of a sequence in a way which allows the axiom of choice to be applied easily.
A collection ℱ of subsets of a set S is a filter if
F1 if F ∈ ℱ and G ⊇ F then G ∈ ℱ,
F2 if F ∈ ℱ and G ∈ ℱ then F ⋂ G ∈ ℱ,
F3 ø ∉ ℱ.
Here are three examples.
• If A is a non-empty subset of S then {F: A ⊆ F} is a filter.
• Suppose that (X, τ) is a topological space, and that x ∈ X. The collection Nx of neighbourhoods of x is a filter.
• If (sn) is a sequence in S then
is a filter.
Filters can be ordered. We say that G refines ℱ, and write G ≥ ℱ, if G ⊇ ℱ.
We now consider a topological space (X, τ). We say that a filter ℱ converges to a limit x (and write ℱ → x) if ℱ refines Nx. Clearly if G refines ℱ and ℱ → x then G → x.
The Hausdorff property can be characterized in terms of convergent filters.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.