G-convergences and g-sequential spaces for g-methods
G-methodlar için g-yakınsaklık ve g-dizisel uzaylar
- Tez No: 963552
- Danışmanlar: PROF. DR. OSMAN MUCUK
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2025
- Dil: İngilizce
- Üniversite: Erciyes Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Matematik Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 84
Özet
Bu tezde, k\“{u}meler ve uzaylar \”{u}zerinde tan{\i}ml{\i} genelle\c{s}tirilmi\c{s} yak{\i}nsakl{\i}k yap{\i}lar{\i} olan $G$-methodlar incelenmi\c{s}tir. {\.I}lk olarak, dizilerin yak{\i}nsakl{\i}\u{g}{\i}n{\i}n genellemesi olan $G$-yak{\i}nsakl{\i}k ve $G$-dizisel yak{\i}nsakl{\i}k kavramlar{\i} tan{\i}mlanm{\i}\c{s}t{\i}r. \c{C}al{\i}\c{s}man{\i}n amac{\i}, $G$-yak{\i}nsakl{\i}k ile $G$-dizisel yak{\i}nsakl{\i}k aras{\i}ndaki farklar{\i} ortaya koymakt{\i}r. $G$-yak{\i}nsakl{\i}k yaln{\i}zca diziler \“{u}zerinde tan{\i}ml{\i} fonksiyonel bir metoda dayan{\i}rken, $G$-dizisel yak{\i}nsakl{\i}k $G$-a\c{c}{\i}k kom\c{s}uluk yap{\i}lar{\i} ile tan{\i}mlanarak daha topolojik anlaml{\i} bir yakla\c{s}{\i}m sunar. Bu durum \”{o}rneklerle a\c{c}{\i}k\c{c}a g\“{o}sterilmi\c{s}tir. Ayr{\i}ca, $G$-dizisel a\c{c}{\i}k k\”{u}melerin bir topoloji olu\c{s}turdu\u{g}u ve bunun da $G$-dizisel uzay olarak isimlendirildi\u{g}i ispatlanm{\i}\c{s}t{\i}r. Ancak, $G$-a\c{c}{\i}k k\“{u}meler ailesinin b\”{o}yle bir topoloji olu\c{s}turmad{\i}\u{g}{\i} g\“{o}r\”{u}lm\“{u}\c{s}t\”{u}r. Sonras{\i}nda, bu fikirler geni\c{s}letilerek $G$-yak{\i}nsakl{\i}k ve $G$-dizisel uzaylar ba\u{g}lam{\i}nda $G$-submethod kavramlar{\i} tan{\i}mlanm{\i}\c{s} ve ara\c{s}t{\i}r{\i}lm{\i}\c{s}t{\i}r. Tezin devam{\i}nda, $G$-method ba\u{g}lam{\i}nda s\“{u}reklilik kavram{\i} incelenmi\c{s}tir. Bu kapsamda, $G$-s\”{u}reklilik ve $G$-dizisel s\“{u}reklilik tan{\i}mlanm{\i}\c{s}, ard{\i}ndan farkl{\i} $G$ ve $H$ methodlar{\i} aras{\i}nda tan{\i}mlanan $(G,H)$-s\”{u}reklilik ve $(G,H)$-dizisel s\“{u}reklilik kavramlar{\i}na ge\c{c}ilmi\c{s}tir. $G$-s\”{u}reklilik ile $G$-dizisel s\“{u}reklilik aras{\i}ndaki temel farklar ortaya konulmu\c{s} ve bu farklar \c{c}e\c{s}itli kar\c{s}{\i} \”{o}rneklerle desteklenmi\c{s}tir. Tezin son b\“{o}l\”{u}m\“{u}nde, $G$-yak{\i}nsak ayr{\i}\c{s}{\i}m aksiyomlar{\i} ($G$-$\mathcal{T}_i$, $i = 0,1,2,3,4$) tan{\i}t{\i}lm{\i}\c{s} ve incelenmi\c{s}tir. Bu aksiyomlar, $G$-a\c{c}{\i}k ve $G$-kapal{\i} altk\”{u}meler kullan{\i}larak karakterize edilmi\c{s} ve elde edilen sonu\c{c}lar a\c{c}{\i}klay{\i}c{\i} \"{o}rneklerle desteklenmi\c{s}tir.
Özet (Çeviri)
In this thesis, we study generalized convergence structures known as $G$-methods on sets and spaces. First, we investigate two types of convergence, $G$-convergence and $G$-sequential convergence. The study aims to find the differentiation between G-convergence and G-sequential convergence. While G-convergence relies purely on the functional method applied to sequences, G-sequential convergence incorporates $G$-open neighborhood structures and offers a more topologically meaningful approach. We demonstrate thorough counterexamples that these notions do not generally coincide. We prove that the family of G-sequentially open sets forms a topology known as $G$-sequential space, unlike the family of G-open sets. Subsequently, we extend these ideas to define and explore $G$-submethods in $G$-convergence and in $G$-sequential spaces. We then analyze continuity in the context of $G$-methods, introducing notions of $G$-continuity, $G$-sequential continuity. Then study advances to the concept of $(G,H)$-continuity and $(G,H)$-sequential continuity, introducing mappings between distinct $G$ and $H$ methods. Important differences between $G$-continuity and $G$-sequential continuity are discussed, and different counterexamples support the idea. The last chapter of the thesis introduces and studies $G$-convergently separation axioms ($G$-$T_i$, $i=0,1,2,3,4$). We characterize these axioms using $G$-open and $G$-closed subsets, and supported our results through illustrative counterexamples. While rooted in pure mathematics, this research is also motivated by practical considerations. In the future, I would like to connect these concepts with computation and data-driven theory, especially in areas such as decision-making, data analysis, and algorithm design. This reflects the general idea of transforming theoretical foundations into practical tools for addressing real-world problems.
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