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Undergraduate Texts in Mathematics (UTM) (ISSN 0172-6056) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag.The books in this series, like the other Springer-Verlag mathematics series, are small yellow books of a standard size.
For example, the Euclidean topology on the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set. But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism , though usually most classify up to homotopy equivalence .
The product topology on is equal to the initial topology induced by the canonical projections : as ranges over . [4] Consequently, the initial topology on induced by {:} is equal to the inverse image of the product topology on by the evaluation map:. [4] Furthermore, if the maps {} separate points on then the evaluation map is a homeomorphism ...
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
120 A Basic Course in Partial Differential Equations, Qing Han (2011, ISBN 978-0-8218-5255-2) 121 A Course in Minimal Surfaces, Tobias Holck Colding, William P. Minicozzi II (2011, ISBN 978-0-8218-5323-8) 122 Algebraic Groups and Differential Galois Theory, Teresa Crespo, Zbigniew Hajto (2011, ISBN 978-0-8218-5318-4)
As a less trivial example, consider the space of all rational numbers with their ordinary topology, and the set of all positive rational numbers whose square is bigger than 2. Using the fact that 2 {\displaystyle {\sqrt {2}}} is not in Q , {\displaystyle \mathbb {Q} ,} one can show quite easily that A {\displaystyle A} is a clopen subset of Q ...
Cauchy space – Concept in general topology and analysis; Convergence space – Generalization of the notion of convergence that is found in general topology; Filters in topology – Use of filters to describe and characterize all basic topological notions and results. Sequential space – Topological space characterized by sequences