A topological space X, is shown to be compact if and only if every net in X has a cluster point. If s is a net in a product Q 2A X, where each Xis a compact topological space, then, for every subset B of A, such that the restriction of s to B has a cluster point in the partial product Q 2B X, it is found that the restriction of s to B [ fg – extending B by one element 2 A n B – has a cluster point in its respective partial product Q 2B[fg X, as well. By invoking Zorn’s lemma, the whole of s can be shown to have a cluster point. It follows that the product of any family of compact topological spaces is compact with respect to the product topology. This is Tychono’s theorem. The aim of this text is to set forth a self contained presentation of this proof. Extra attention is given to highlight the deep dependency on the axiom of choice.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:oru-44729 |
| Date | January 2015 |
| Creators | Hörngren, Gustav |
| Publisher | Örebro universitet, Institutionen för naturvetenskap och teknik |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Student thesis, info:eu-repo/semantics/bachelorThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
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