THE BOLZANO-WEIERSTRASS THEOREM AND METRIC SPACE
Keywords:
Metric Space, Distance FunctionAbstract
Numerous mathematical structures, including the well-known Euclidean spaces and others, may be represented using metric spaces. Metric spaces are fundamental to the mathematical study of continuity, convergence, and limiting behaviour. It lays the groundwork for defining and evaluating ideas like open and closed sets, function continuity, and sequence and series convergence. Metric spaces are useful in many areas of mathematics, such as real analysis, functional analysis, topology, and more, since they use the idea of distance to facilitate the manipulation of a wide variety of mathematical objects. Metric spaces are fundamental to mathematics because they provide a systematic framework for studying the attributes and connections between points.
References
C. Blake and C. Merz. UCI repository of machine learning databases, 1998.
M. DeGroot and S. Fienberg. The comparison and evaluation of forecasters. Sttistician,32(1):12--22, 1982.
P. Giudici. Applied Data Mining. John Wiley and Sons, New York, 2003. Gualtieri, S. R. Chettri, R. Cromp, and L. Johnson. Support vector machine classifiers as applied to aviris data. In Proc. Eighth JPL Airborne Geoscience Workshop, 1999.
T. Joachims. Making large-scale svm learning practical. In Advances in Kernel Methods, 1999.
R. King, C. Feng, and A. Shutherland. Statlog: comparison of classification algorithms on large real-world problems. Applied Artificial Intelligence, 9(3):259--287, May/June 1995.
P.A. Flach. The geometry of roc space: understanding machine learning metrics through roc isometrics. In Proc. 20th International Conference on Machine Learning (ICML'03), pages 194--201. AAAI Press, January 2003.
