Eksponenti topoloških prostorov : delo diplomskega seminarja

Reberc, Petra Ivana (author), Bauer, Andrej (mentor), Lešnik, Davorin (comentor)

Zanima nas, kdaj lahko na prostoru zveznih funkcij $C(X, Y)$ vpeljemo eksponentno topologijo. Ugotovimo, da je to odvisno predvsem od lastnosti prostora $X$ oziroma domene, in z obravnavanjem prostora Sierpińskega kot kodomeno poenostavimo problem. Raziskujemo topologije na družinah odprtih množic in se tako približujemo iskanemu pogoju. Ugotovimo, da mora biti $X$ sržno kompakten in podamo eksponentno topologijo za dani $X$.

• Language: Slovenian
• Keywords: eksponentna topologija, eskponenciabilnost, sržno kompaktni prostori, šibka topologija, krepka topologija, Scottova topologija, topologija Aleksandrova
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-103478
• UDC: 515.1
• COBISS.SI-ID: 18441305
• Publication date in RUL: 19.09.2018

Secondary language

• Language: English
• Title: Exponentiable spaces and the exponential topology

Abstract

We would like to know when is it possible to endow $C(X, Y)$ with an exponential topology. We observe this depends mainly on the properties of the domain $X$ and then we simplify the problem by considering the Sierpiński space as the codomain. We explore topologies on topologies and approach the desired criteria. We find out $X$ has to be core compact and find the adequate exponential topology.

• Keywords: exponential topology, exponentiability, core-compact spaces, weak topology, strong topology, Scott topology, Alexandroff topology