Editing Tychonoff's theorem (section)

{{Short description|Product of any collection of compact topological spaces is compact}}
{{For|other theorems named after Tychonoff|Tychonoff's theorem (disambiguation)}}
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In [[mathematics]], '''Tychonoff's theorem''' states that the product of any collection of [[compact space|compact]] [[topological space]]s is compact with respect to the [[product topology]]. The theorem is named after [[Andrey Nikolayevich Tikhonov]] (whose surname sometimes is transcribed ''Tychonoff''), who proved it first in 1930 for powers of the closed [[unit interval]] and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article by Tychonoff, {{lang|de|italic=no|"Über einen Funktionenraum"}}.<ref>{{cite journal |last=Tikhonov |first=Andrey Nikolayevich |title=Über einen Funktionraum |language=de |journal=Mathematische Annalen |issue=111 |pages=762–766 |year=1935}}</ref>

Tychonoff's theorem is often considered as perhaps the single most important result in general topology (along with [[Urysohn's lemma]]).<ref>{{cite book |last=Willard |first=Stephen |author-link=Stephen Willard |title=General Topology |year=2004 |publisher=Dover |isbn=978-0-486-43479-7 |page=120}}</ref> The theorem is also valid for topological spaces based on [[fuzzy sets]].<ref>{{cite journal |last=Goguen |first=Joseph |author-link=Joseph Goguen |title=The Fuzzy Tychonoff Theorem |journal=[[Journal of Mathematical Analysis and Applications]] |volume=43 |issue=3 |date=September 1973 |pages=734–742|doi=10.1016/0022-247X(73)90288-6 }}</ref>