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.

The significance of search engine optimization is clearly highlighted sin- ce over 90 % of internet usage starts with a search engine according to contemporary studies, whilst the internet also has contributed to increa- sed competition between companies operating from different geographic markets. Guidelines from search engines and theory from studies descri- be several internal influencing factors, on-site factors, but ambiguous in the studies is how much these factors affect the ranking. For this reason, a study was conducted regarding the e-commerce platform Sparket.se and its ranking on Google through iterative testing and measurement of each on-site factor. The result and conclusion of the study suggest that key- words and structure are of greatest importance in on-site optimization. The largest improvement was observed when keywords were placed in the HTML code and accounted for 2-5 % of the total content, without duplication. A clear hierarchical structure of even width and depth also resulted in an advancement. Implementation of breadcrumbs again pro- vided limited improvement, likely because of Sparket’s small size, and sitemap, as well as robots.txt, yielded unchanged ranking. The overall effect of the on-site factors in search engine optimization was significant for the e-commerce platform. / Vikten av sökmotoroptimering belyses tydligt eftersom över 90 % av in- ternetanvändningen inleds med en sökmotor enligt nutida studier, sam- tidigt som internet har bidragit till ökad konkurrens mellan företag som verkar från olika geografiska marknader. Riktlinjer från sökmotorer och teori från studier beskriver flera interna påverkande faktorer, on-site- faktorer, men tvetydigt i studierna är hur mycket dessa faktorer påver- kar rangordningen. Med anledning av detta genomfördes en studie an- gående e-handelsplattformen Sparket.se och dess rangordning på Goog- le genom iterativ testning och mätning för varje on-site-faktor. Resulta- tet och slutsatsen av arbetet gör gällande att nyckelord och struktur har störst betydelse vid on-site-optimering. Störst förbättring observerades när nyckelord placerades i HTML-koden och utgjorde 2-5 % av det tota- la innehållet, utan duplicering. En tydlig hierarkisk struktur med jämn bredd och djup resulterade likaså i avancemang. Implementation av bre- adcrumbs gav också en marginell förbättring, troligtvis på grund av Spar- kets ringa storlek, och sitemap samt robots.txt gav oförändrad rangord- ning. Den sammanlagda effekten av on-site-faktorerna vid sökmotorop- timering var signifikant för e-handelsplattformen.