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Inverzne limite inverznih zaporedij z navzgor polzveznimi večličnimi veznimi preslikavami : doktorska disertacijaBanič, IztokV disertaciji bomo najprej preučevali inverzne limite inverznih zaporedij enotskih intervalov ▫$\lbrack 0,1\rbrack$▫, vendar bomo namesto enoličnih veznih funkcij na ▫$\lbrack 0,1\rbrack$▫ ... obravnavali navzgor polzvezne večlične funkcije, ki jih dobimo iz danih enoličnih zveznih funkcij na ▫$\lbrack 0,1\rbrack$▫ s posebnim standardnim postopkom, imenovali jih bomo kontinuumi z jedrom. Podali bomo zglede in dokazali zanimive lastnosti takih inverznih limit, na primer: 1. Jedro je konvergenčni kontinuum v kontinuumu z jedrom. 2. Jedro kontinuuma z jedrom je limita lokov glede na Hausdorffovo metriko v ustreznem hiperprostoru. 3. Za vsak kontinuum ▫$X$▫ z jedrom ▫$K$ obstaja družina lokov ▫$\lbrace L_{\alpha}\vert \alpha \in A \rbrace$▫, tako da je ▫$X = K \cup (\bigcup _{\alpha \in A} \L_\alpha). Dokazali bomo tudi, da pri določenih pogojih velja sklep iz Mahavierjeve domneve, ki pravi, da je za vsako navzgor polzvezno večlično funkcijo ▫$f : \lbrack O,1 \rbrack \to \lbrack 0,1 \rbrack$▫ dimenzija inverzne limite ▫$\underline {lim} \lbrace \lbrack 0,1 \rbrack , f \rbrace _{n=1^\infty}$▫ enaka bodisi ▫$1$▫ bodisi ▫$ \infty $▫. Predstavili bomo tudi splošnejši primer, kako dobiti zanimive primere inverznih limit inverznih zaporedij poljubnih kompaktnih metričnih prostorov ▫$X_n$▫ in navzgor polzveznih večličnih funkcij ▫$ \tilde {f}_{n} : X_{n+1} \to X_n $▫ iz podanih enoličnihzveznih funkcij ▫$f_n$▫. Dokazali bomo izreke o dimenziji takih inverznih limit: 1. Naj bo ▫$K$▫ inverzna limita inverznega zaporedja kompaktnih metričnih prostorov ▫$X$▫ in zveznih preslikav ▫$ f_n : X \to X $▫ in naj bo za vsako naravno število ▫$n$▫, ▫$A_n$▫ zaprta podmnožica ▫$X$▫. Tedaj bodisi obstaja celo število ▫$m \geq, \ge 0$▫, tako da je ▫$dim(\tilde{K}) = dim(D_{m} x X)$▫ bodisi je ▫$ dim(\tilde{K}) = \infty$▫. 2. Naj bo ▫$X$▫ nedegeneriran kompakten metrični prostor, ▫$A$▫ zaprta podmnožica prostora ▫$X$▫ in ▫$f : X \to X$▫ zvezna preslikava. Naj bo nadalje ▫$K = \underline{lim} \lbrace X,f \rbrace _{n=1^\infty}$▫. Tedaj je dimenzija prostora ▫$\tilde{K}$▫ enaka bodisi ▫dim(X)$▫ bodisi ▫$\infty$▫. Na koncu bomo dokazali še, da je inverzna limita poljubnega inverznega zaporedja kompaktnih metričnih prostorov in surjektivnih veznih preslikav enaka limiti ustrezno izbranih homeomorfnih kopij istih prostorov v ustreznem hiperprostoru, glede na Hausdorffovo metriko.Type of material - dissertation ; adult, seriousPublication and manufacture - [Maribor : I. Banič], 2007Language - slovenianCOBISS.SI-ID - 15459848
Author
Banič, Iztok
Other authors
Milutinović, Uroš
Topics
topologija |
kontinuum |
nerazcepnost |
topološka dimenzija |
inverzno zaporedje |
inverzna limita |
polzvezna funkcija |
n-drevo |
disertacije
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Banič, Iztok | 23201 |
Milutinović, Uroš | 08727 |
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