General topology

It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.

The notation Xτ may be used to denote a set X endowed with the particular topology τ.

For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.

This example shows that in general topological spaces, limits of sequences need not be unique.

This gives back the above δ-ε definition of continuity in the context of metric spaces.

Note, however, that if the target space is Hausdorff, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a).

In several contexts, the topology of a space is conveniently specified in terms of limit points.

This motivates the consideration of nets instead of sequences in general topological spaces.

If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

A topology on a set S is uniquely determined by the class of all continuous functions

into all topological spaces X. Dually, a similar idea can be applied to maps

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

Explicitly, this means that for every arbitrary collection of open subsets of X such that there is a finite subset J of A such that Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets.

Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.

For a topological space X the following conditions are equivalent: Every interval in R is connected.

It follows that, in the case where their number is finite, each component is also an open subset.

The space X is said to be path-connected (or pathwise connected or 0-connected) if there is at most one path-component; that is, if there is a path joining any two points in X.

The converse is not always true: examples of connected spaces that are not path-connected include the extended long line L* and the topologist's sine curve.

A subset of X is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form pi−1(U).

The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.

Continuum theory is the branch of topology devoted to the study of continua.

These objects arise frequently in nearly all areas of topology and analysis, and their properties are strong enough to yield many 'geometric' features.

Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change.

Many examples with applications to physics and other areas of math include fluid dynamics, billiards and flows on manifolds.

The topological characteristics of fractals in fractal geometry, of Julia sets and the Mandelbrot set arising in complex dynamics, and of attractors in differential equations are often critical to understanding these systems.

The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).

It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC).

The answer to the normal Moore space question was eventually proved to be independent of ZFC.

Some standard books on general topology include: The arXiv subject code is math.GN.