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"ABSTRAKSkripsi ini membahas mengenai salah satu sifat topologi yaitu ruang tipis. Pada penulisan skripsi ini akan dibahas selain definisi dari ruang tipis akan ditunjukan sebuah syarat cukup agar sebuah ruang metrik dapat dikatakan tipis. Selain membahas syarat cukup dari sebuah ruang metrik agar dapat dikatakan tipis akan dibahas juga keterkaitan antara ruang metrik yang uniform approachable dengan ruang tipis. Setelah itu akan dibahas juga mengenai salah satu sifat yang dimiliki oleh ruang metrik yang tipis yaitu sifat pemisahan kompaknya.

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ABSTRACT

This undergraduate thesis discusses about one of the topology property which is thin space. This undergraduate thesis will show that there is one requirement to show that a metric space is thin or not beside the original definition. Beside that this undergraduate thesis will also discusses relationship between a uniform approachable metric space and thin space. After that this undergraduate thesis will also discusses about one of the properties that from a thin metric space have which is the compact separation property."

"Dalam analisis riil, atau lebih luas dalam ruang metrik, dikenal konsep pada fungsi kontinu. adalah suatu konstanta positif yang bergantung pada dan, dengan anggota domain dari fungsi kontinu tersebut. Pada ruang metrik, bilangan tidak hanya sebuah bilangan yang bergantung pada dan tetapi dapat merupakan sebuah fungsi kontinu. Dalam skripsi ini, dipelajari konstruksi sehingga menjadi suatu fungsi kontinu yang bergantung pada dan . Lebih lanjut, diperlihatkan bahwa terdapat sifat ruang metrik yang mempengaruhi sifat fungsi yakni nilai minimum pada fungsi di ruang metrik yang sequentially compact."

"Titik x dikatakan titik tetap dari sembarang pemetaan T jika dan hanya jika T x = x. Berbagai hasil mengenai teorema titik tetap telah dibuktikan pada ruang metrik. Seiring berkembangnya bidang analisis matematis, semakin banyak matematikawan yang berhasil membuktikan teorema titik tetap di berbagai ruang dan pemetaan. Namun, tidak banyak hasil mengenai teorema titik tetap yang telah dibuktikan pada ruang dislocated quasi b-metric. Ruang dislocated quasi b-metric adalah salah satu bentuk perluasan dari ruang metrik dimana jarak antara dua buah titik yang sama tidak harus bernilai nol yaitu d(x, x) =/= 0 serta sifat simetri yaitu d(x, y) = d(y, x) tidak berlaku di ruang ini. Pada skripsi ini, akan dibuktikan kembali teorema-teorema mengenai ketunggalan titik tetap pada ruang dislocated quasi b-metric untuk sembarang pemetaan. Pada Skripsi ini juga akan dibahas mengenai teorema titik tetap untuk pemetaan tipe F-kontraktif pada ruang yang sama.

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A point x is said to be a fixed point of a mapping T on a nonempty set X if and only if T x = x. Many results regarding the fixed point theorem have been proved on metric spaces. As the field of mathematical analysis develops, more and more mathematicians have succeeded in proving fixed point theorems in various spaces and mappings. However, not many results regarding the fixed point theorem have been proved on dislocated quasi b-metric spaces. Dislocated quasi b-metric space is one of the extensions of metric space where the distance between two equal points does not have to be zero i.e. d(x, x) =/= 0 and the symmetry property i.e. d(x, y) = d(y, x) does not apply in this space. In this thesis, we will prove the theorems on the uniqueness of fixed points on dislocated quasi b-metric spaces for any mapping. This thesis also discusses the fixed point theorem for F-contractive type mappings in the same space."

"Suatu ruang metrik disebut lengkap apabila setiap barisan Cauchy di ruang metrik tersebut adalah konvergen. Ruang Atsuji adalah ruang metrik lengkap di mana setiap fungsi kontinu bernilai real adalah kontinu seragam. Suatu ruang metrik dikatakan memiliki Atsuji completion jika completion dari ruang metrik tersebut adalah ruang Atsuji. Dalam makalah ini dipelajari sifat subhimpunan di ruang metrik yang completion-nya adalah ruang Atsuji, yaitu sifat subhimpunan lengkap dan sifat subhimpunan completely discrete.

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Complete metric space is a metric space where every Cauchy sequence is convergent. Atsuji space is a complete metric space where every real valued and continuous function on it is uniformly countinuous. A metric space is said to have an Atsuji Completion if its completion is an Atsuji space. In this paper, the properties of subsets of a metric space that has an Atsuji completion is explored. The subsets being discussed are complete subsets and completely discrete subsets."

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ABSTRACTAs part of the society, woman who lives in urban slum areas usually did not get much attention of their existence. When we talk about their roles in their life, actually these women have three roles. These three roles are not only reproductive role or dosmectic role, when women should take care of their family and house, but also productive role or the role that makes them to work too as a primer or secondary income-earners, and the third role, the community management (Moser, 1988; 1993, dalam Miraftab, 1995).These women in slum areas became special as their three roles should meet the economic presure and bad condition of their settelement. They should face the truth about patriarchy system in the family, which men (or their husband) is said to have domination, including women space of activities.From the pre-research in 2012, researcher found out that women in slum areas can use even the small space in their house efficiently and effectively. But at that pre-research, researcher did not include about the women point of view of the use of that space with their husband or children. Because of that, this research is about exploring women’s life and apce of activities in slum areas with their roles and activities in spaces in slum areas from gender perspective of how the space is created and used as it related with people around the women’s life, especially their husband, when they use that space.The case studies of this reasearch are slum areas in Bukit Duri, Tebet, South Jakarta, with focus on RT 11 and RT 15 in RW 10. Both RTs is the poorest and always got flooded when the rain season comes because of their location in Kali Ciliwung riverbank. These women who live in this settlements can show us that poor women can live their life in very bad and limited spaces in their settlement for their space of activities. The limited spaces meet a lot of needs of space then relates to the existence of negotiation that women did as the way they form and use the existing spaces for their space of activities.The method that researcher used is ethnograpic method, especially about the relation of people and spaces. The main approach is participant-observation to know these women’s space, people who involved, activities, the objects, actions, events, aims, and feeling in each of their activities (Reeves, Kuper, dan Hodges, 2008). With that method, research can learn how these women and their family, especially their husband, use the space in their house and their society for their activities as part of space negotiation.Research then found out that women in slum areas can balance their three roles and that’s relate to how their face life presure from their husband, children, and the condition of their house and settlement. But at the same time, all of that pressures can make them stronger and can get opportunity to have access of spaces in their house and settlement. This can show us the position of these women in family are not subordinated. Even that position can be seen in how they use the spaces for their activities as a result of space negotiation that women and their husband did. So all of the condition that being thought will make them have a hard time to do their activities is actually can help them balance their roles, which area reproductive role, productive role, and community management role."