# Brief Introduction to Topology

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Topology is one of four major areas of abstract mathematics: algebra (equations), analysis (limits), foundations (set theory and logic), and topology. The word topology comes from the Greek word for place: .
Broadly speaking, topology is the study of space and continuity. Since topology includes the study of continuous deformations of a space, it is often popularly called rubber sheet geometry. Vector spaces Rn are spaces, but a space can be a finite set of points or an infinite dimensional space of functions. The surfaces of a donut and a sphere appear the same on a small scale, so what properties distinguish them? To distinguish various spaces, notions from algebra or analysis are often introduced. We may associate groups, rings, modules, or other algebraic objects to a topological space in order to precisely describe information. These constructions may even give information back about algebra, e.g., group cohomology, or analysis, e.g., de Rham cohomology.
At LSU, topologists study a variety of topics such as spaces from algebraic geometry, topological semigroups and ties with mathematical physics. A major topic studied at LSU is the placement problem. This problem is to determine the manner in which a space N can sit inside of a space M. Usually there is some notion of equivalence. If N is the circle and M is R3 then the subject is classical knot theory. A map of the circle into R3 can be the obvious one we can draw on paper or a complicated one which is knotted and can't be drawn on paper without crossing itself. Studying the placements of circles in 3-dimensional manifolds can inform us about the manifold. Other versions of the placement problem that have been or are presently studied at LSU are hyperplanes in a vector space, graphs in R3, and n-dimensional spheres in (n+2)-dimensional spheres.