Topology

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Contents

[edit] Introduction

General Topology is based solely on set theory and concerns itself with structures of sets. It is at its core a generalization of the concept of distance, though this will not be immediately apparent for the novice student. Topology generalizes many distance related concepts, such as continuity, compactness and convergence.

For an overview of the subject of topology, please see the Wikipedia entry.

[edit] Before You Begin

In order to make things easier for you as a reader, as well as for the writers, you will be expected to be familiar with a few topics before beginning.

[edit] Point - Set Topology

[edit] Some Set Theory

[edit] Introduction to Topology

[edit] Properties of Topological Spaces


[edit] Algebraic Topology

[edit] Homotopy

[edit] Polytopes

[edit] Homology

[edit] Differential Topology

[edit] Help

[edit] Question & Answer

Have a question? Why not ask the very textbook that you are learning from?


1. What is the difference between topology, algebra and analysis?

  • Topology is a generalization of analysis and geometry. It comes in two main flavors: point-set topology and algebraic topology. The former generalizes concepts from analysis dealing with space such as continuity of functions, connectedness of a space, open and closed sets, (etc.). The latter attributes algebraic structures (groups, rings etc.) to families of topological spaces to distinguish topological differences in those families. A naive description of topology is that it identifies those qualities of a space that do not change under twisting and stretching of that space. As such, it is popularly referred to as "rubber sheet geometry." In reality topology does far more than this, in fact providing a rigorous foundation under all branches of mathematics dealing with "spaces." Differential topology is a sort of hybrid between algebraic topology and geometry, where distance is still irrelevant, but "smoothness" of functions and surfaces is detectable (and required).
  • Algebra deals with binary operations on sets, where two elements are combined to make one (such as sums and products). Major areas of interest in algebra are group theory, ring theory and field theory. Each of these focuses on the cases where the operation obeys their respective qualities.
  • Analysis (or specifically real analysis) on the other hand deals with the real numbers \mathbb{R} and the standard topology and algebraic structure of \mathbb{R}.


2. How are the concepts of base and open cover related? It seems that every base is an open cover, but not every open cover is a base. But, why are both concepts needed?

  • The terms base and open cover are not evidently related. Every base is an open cover which is probably the main relation. Take a second countable topological space for instance (second countable means that the space has a countable base for its topology). Such a space satisfies the property that every open cover has a countable sub-cover. To prove this we use the countability of the base. Basically, for any open cover, we choose for each element of the space, an element of the open cover containing it and hence a basis element contained in that element of open cover. Therefore, for any open cover, we can generate a open cover of basis elements that is an 'open refinement' (see Wikipedia for definition). From here we can get properties of open covers from properties of the base. If the base is countable, we can generate a countable open cover from the original cover. Does this answer you question?

Topology Expert (talk) 04:17, 8 June 2008 (UTC)

[edit] Further Reading

[edit] General Topology

Aleksandrov; Combinatorial Topology (1956)

Baker; Introduction to Topology (1991)

Dixmier; General Topology (1984)

Engelking; General Topology (1977)

Munkres; Topology (2000)

James; Topological and Uniform Spaces (1987)

Jänich; Topology (1984)

Kuratowski; Introduction to Set Theory and Topology (1961)

Kuratowski; Topology (1966)

Roseman; Elementary Topology (1999)

Seebach, Steen; Counterexamples in Topology (1978)

Willard; General Topology (1970)

[edit] Algebraic Topology

Marvin Greenberg and John Harper; Algebraic Topology (1981)

Allen Hatcher, Algebraic Topology (2002) [1]

Hu, Sze-tsen, Cohomology Theory (1968)

Hu, Sze-tsen, Homology Theory (1966)

Hu, Sze-tsen, Homotopy Theory (1959)

Albert T. Lundell and Stephen Weingram, The Topology of CW Complexes (1969)

Joerg Mayer, Algebraic Topology (1972)

James Munkres, Elements of Algebraic Topology (1984)

Joseph J. Rotman, An Introduction to Algebraic Topology (1988)

Edwin Spanier, Algebraic Topology (1966)

[edit] External Links

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