The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.

Some theorems related to compactness ( see the glossary of topology for the definitions ):

Differential geometry is closely related to differential topology, and to the geometric aspects of the theory of differential equations.

The emergence of the code is governed by the topology defined by the probable errors and is related to the map coloring problem.

In condensed matter physics, the three dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms.

In topology and related branches of mathematics, a Hausdorff space, separated space or T < sub > 2 </ sub > space is a topological space in which distinct points have disjoint neighbourhoods.

In mathematics, more specifically in general topology and related branches, a net or Moore – Smith sequence is a generalization of the notion of a sequence.

The ideas of pointless topology are closely related to mereotopologies in which regions ( sets ) are treated as foundational without explicit reference to underlying point sets.

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

* Subspace topology, in topology and related areas of mathematics

In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T < sub > 4 </ sub >: every two disjoint closed sets of X have disjoint open neighborhoods.

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider.

In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods.

As of August, 2011, he has more than 340 publications primarily in quantum field theory and string theory and in related areas of topology and geometry.

In topology and related branches of mathematics, a topological space X is a T < sub > 0 </ sub > space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.

Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects ( for example, numbers and functions ) from all the traditional areas of mathematics ( such as algebra, analysis and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them.

Both these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.

Real analysis is also used as a starting point for other areas of analysis, such as complex analysis, functional analysis, and harmonic analysis, as well as for motivating the development of topology, and as a tool in other areas, such as applied mathematics.

Although there is no absolute distinction between different areas of topology, the focus here is on general topology.

Other " miscellaneous " research areas in theoretical chemistry include the mathematical characterization of bulk chemistry in various phases ( e. g. the study of chemical kinetics ) and the study of the applicability of more recent math developments to the basic areas of study ( e. g. for instance the possible application of principles of topology to the study of elaborate electronic structure ).

While he is best known for the Kolmogorov – Arnold – Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

The logical view is that OSPF creates something of a spider web or star topology of many areas all attached directly to Area Zero and IS-IS by contrast creates a logical topology of a backbone of Level 2 routers with branches of Level 1-2 and Level 1 routers forming the individual areas.

Nineteen barrios comprise the rural areas of the municipality, and the topology of their lands varies from flatlands to hills to steep mountain slopes.

General topology grew out of a number of areas, most importantly the following:

As the name implies, general topology provides the common foundation for these areas.

What was really eye-catching, from the point of view of other mathematical areas, was the proposed connection with algebraic topology.

* Atlas ( topology ), a collection of local coordinate charts in mathematics

It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.

In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

Dyson also did work in a variety of topics in mathematics, such as topology, analysis, number theory and random matrices.

In mathematics, graphs are useful in geometry and certain parts of topology, e. g. Knot Theory.

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.

Likewise, analysis, geometry and topology, although considered pure mathematics, find applications in theoretical physics — string theory, for instance.

Another aspect of mathematics, set-theoretic topology and point-set topology, concerns objects of a different nature from objects in our universe, or in a higher dimensional analogue of our universe.

The mathematician Leonhard Euler was one of the first to analyze plane mazes mathematically, and in doing so made the first significant contributions to the branch of mathematics known as topology.

In mathematics, pointless topology ( also called point-free or pointfree topology ) is an approach to topology that avoids mentioning points.