** On every infinite-dimensional topological vector space there is a discontinuous linear map.

** In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism ( see homeomorphism group ).

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

* Base ( topology ), a topological space

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

Basic constructions, such as the fundamental group or fundamental groupoid of a topological space, can be expressed as fundamental functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications.

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

In general topological spaces, however, the different notions of compactness are not equivalent, and the most useful notion of compactness — originally called bicompactness — involves families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

Specifically, a topological space is compact if, whenever a collection of open sets covers the space, some subcollection consisting only of finitely many open sets also covers the space.

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

A subset K of a topological space X is called compact if it is compact in the induced topology.

Ultimately the Russian school of point-set topology, under the direction of Pavel Alexandrov and Pavel Urysohn, formulated Heine – Borel compactness in a way that could be applied to the modern notion of a topological space.

* Any finite topological space, including the empty set, is compact.

Then is a compact topological space ; this follows from the Tychonoff theorem.

* Every topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.

# A topological space X is compact.

* A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.

An embedding of a topological space X as a dense subset of a compact space is called a compactification of X.

For any topological space X the ( Alexandroff ) one-point compactification αX of X is obtained by adding one extra point ∞ ( often called a point at infinity ) and defining the open sets of the new space to be the open sets of X together with the sets of the form G < font face =" Arial, Helvetica "> U </ font >

If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.

If X is a topological space, there is a natural way of transforming X /~ into a topological space ; see quotient space for the details.

Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open ( X ) under inclusion.

The objects are pairs ( X, x < sub > 0 </ sub >), where X is a topological space and x < sub > 0 </ sub > is a point in X.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.

Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's that there is a deep connection between the topological characteristics of a variety and its diophantine ( number theoretic ) properties.

The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous, ( and hence homeomorphism ) so that it forms a topological group under multiplication.

The quantum hall effect is another example of measurements with high magnetic fields where topological properties such as Chern-Simons angle can be measured experimentally.

A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps ( morphisms ) between topological spaces in topology ( the associated category is called Top ), and the study of smooth functions ( morphisms ) in manifold theory.

is in C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected.

In mathematics, compactification is the process or result of making a topological space compact.

Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is regular, Hausdorff and second-countable.

It states that a topological space is metrizable if and only if it is regular, Hausdorff and has a σ-locally finite base.

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

A topological space is Tychonoff if and only if it's both completely regular and T < sub > 0 </ sub >.

Almost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular.

* Every topological group is completely regular.

* Generalising both the metric spaces and the topological groups, every uniform space is completely regular.

Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).

Thus the category of completely regular spaces CReg is a reflective subcategory of Top, the category of topological spaces.

As a uniform space, every topological group is completely regular.

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.

A T < sub > 3 </ sub > space or regular Hausdorff space is a topological space that is both regular and a Hausdorff space.

A locally regular space is a topological space where every point has an open neighbourhood that is regular.

There are many results for topological spaces that hold for both regular and Hausdorff spaces.

Most topological spaces studied in mathematical analysis are regular ; in fact, they are usually completely regular, which is a stronger condition.

In fact, this property characterises regular spaces ; if the closed neighbourhoods of each point in a topological space form a local base at that point, then the space must be regular.

This property is actually weaker than regularity ; a topological space whose regular open sets form a base is semiregular.

Every such regular cover is a principal G-bundle, where G = Aut ( p ) is considered as a discrete topological group.

* Free regular set, a subset of a topological space that is acted upon disjointly under a given group action.

It is in this form that the regular representation is generalized to topological groups such as Lie groups.