Basic Properties of Metrizable Topological Spaces

Pąk, Karol

doi:10.2478/v10037-009-0024-8

Cited by 7 publications

(13 citation statements)

References 1 publication

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“…In 1972, a first report was published and presented at UN concerning SD. This report, named as the Meadows Report [1], was strongly criticised at that time since it advocated non-growth to the developing countries [2]. Later in 1987, the Brundtland Report (BR) [3] defined the SD concept as development that meets the essential needs of the present without compromising the ability of future generations to meet their own essential needs.…”

Section: Introductionmentioning

confidence: 99%

Monitoring sustainable development by means of earth observation data and machine learning: a review

2020

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This paper presents and explores the different Earth Observation approaches and their contribution to the achievement of United Nations Sustainable Development Goals. A review on the Sustainable Development concept and its goals is presented followed by Earth Observation approaches relevant to this field, giving special attention to the contribution of Machine Learning methods and algorithms as well as their potential and capabilities to support the achievement of Sustainable Development Goals. Overall, it is observed that Earth Observation plays a key role in monitoring the Sustainable Development Goals given its cost-effectiveness pertaining to data acquisition on all scales and information richness. Despite the success of Machine Learning upon Earth Observation data analysis, it is observed that performance is heavily dependent on the ability to extract and synthesise characteristics from data. Hence, a deeper and effective analysis of the available data is required to identify the strongest features and, hence, the key factors pertaining to Sustainable Development. Overall, this research provides a deeper understanding on the relation between Sustainable Development, Earth Observation and Machine Learning, and how these can support the Sustainable Development of countries and the means to find their correlations. In pursuing the Sustainable Development Goals, given the relevance and growing amount of data generated through Earth Observation, it is concluded that there is an increased need for new methods and techniques strongly suggesting the use of new Machine Learning techniques.

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Section: Introductionmentioning

confidence: 99%

Monitoring sustainable development by means of earth observation data and machine learning: a review

2020

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“…The notation and terminology used in this paper have been introduced in the following articles: [8], [36], [24], [30], [1], [15], [21], [16], [25], [6], [9], [17], [37], [10], [11], [3], [34], [5], [12], [26], [33], [35], [41], [42], [13], [40], [19], [31], [28], [43], [18], [44], [29], and [14].…”

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confidence: 99%

Tietze Extension Theorem for n-dimensional Spaces

Pąk

2014

Formalized Mathematics

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Summary In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of εn with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → εn. Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of En with a non-empty interior. This article is based on [20]; [23] and [22] can also serve as reference books.

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“…The notation and terminology used in this paper have been introduced in the following articles: [16], [1], [13], [12], [11], [14], [19], [18], [9], [2], [10], [3], [7], [20], and [6].…”

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confidence: 99%

Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane

Arnaud¹,

Rudnicki²

2013

Formalized Mathematics

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Summary.We first provide a modified version of the proof in [3] that the Sorgenfrey line is T1. Here, we prove that it is in fact T2, a stronger result. Next, we prove that all subspaces of R 1 (that is the real line with the usual topology) are Lindelöf. We utilize this result in the proof that the Sorgenfrey line is Lindelöf, which is based on the proof found in [8]. Next, we construct the Sorgenfrey plane, as the product topology of the Sorgenfrey line and itself. We prove that the Sorgenfrey plane is not Lindelöf, and therefore the product space of two Lindelöf spaces need not be Lindelöf. Further, we note that the Sorgenfrey line is regular, following from [3]:59. Next, we observe that the Sorgenfrey line is normal since it is both regular and Lindelöf. Finally, we prove that the Sorgenfrey plane is not normal, and hence the product of two normal spaces need not be normal. The proof that the Sorgenfrey plane is not normal and many of the lemmas leading up to this result are modelled after the proof in [3], that the Niemytzki plane is not normal. Information was also gathered from [15].MSC: 54D15 54B10 03B35

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Basic Properties of Metrizable Topological Spaces

Cited by 7 publications

References 1 publication

Monitoring sustainable development by means of earth observation data and machine learning: a review

Monitoring sustainable development by means of earth observation data and machine learning: a review

Tietze Extension Theorem for n-dimensional Spaces

Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane

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