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| Title Details: | |
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Metric spaces |
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| Authors: |
Papadimitrakis, Michail |
| Reviewer: |
Sarantopoulos, Ioannis |
| Subject: | MATHEMATICS AND COMPUTER SCIENCE > MATHEMATICS > REAL FUNCTIONS |
| Description: | |
| Abstract: |
Metrics. Basic examples: mainly the Euclidean spaces and spaces of functions with the metric of uniform convergence. Neighborhoods. Open and closed sets. Interior, closure and boundary of a set. Limits and continuity of functions. Limits of sequences. Complete spaces. Compact spaces. Definition by means of open coverings. Basic properties of compactness. Equivalence with the definition by means of sequences. Compactness in Euclidean spaces: equivalence with the property of being closed and bounded. Continuous real functions on compact sets (boundedness, maximum and minimum values, uniform continuity). Connected spaces. Basic properties of connectedness. Continuous functions on connected sets. Characterization of connected compact sets (by means of finite sequences of points). Arcwise connected sets. Characterization of open connected sets in Euclidean spaces (by means of arcwise connectedness). Connected components.
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| Type: |
Chapter |
| Creation Date: | 2015 |
| Item Details: | |
| License: |
http://creativecommons.org/licenses/by-nc-nd/3.0/gr |
| Handle | http://hdl.handle.net/11419/2901 |
| Bibliographic Reference: | Papadimitrakis, M. (2015). Metric spaces [Chapter]. In Papadimitrakis, M. 2015. Analysis [Undergraduate textbook]. Kallipos, Open Academic Editions. https://hdl.handle.net/11419/2901 |
| Language: |
Greek |
| Is Part of: |
Analysis |
| Publication Origin: |
Kallipos, Open Academic Editions |