In mathematics, and especially general topology, the **Euclidean topology** is the natural topology induced on -dimensional Euclidean space by the Euclidean metric.

In any metric space, the open balls form a base for a topology on that space.^{ [1] } The Euclidean topology on is then simply the topology *generated* by these balls. In other words, the open sets of the Euclidean topology on are given by (arbitrary) unions of the open balls defined as , for all real and all where is the Euclidean metric.

When endowed with this topology, the real line is a T_{5} space. Given two subsets say and of with where denotes the closure of there exist open sets and with and such that ^{ [2] }

In mathematics, a **continuous function** is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be *discontinuous*. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

In mathematical analysis, a metric space M is called **complete** if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.

In mathematics, a **metric space** is a set together with a metric on the set. The metric is a function that defines a concept of *distance* between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In mathematics, **open sets** are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.

In mathematics, the **closure** of a subset *S* of points in a topological space consists of all points in *S* together with all limit points of *S*. The closure of *S* may equivalently be defined as the union of *S* and its boundary, and also as the intersection of all closed sets containing *S*. Intuitively, the closure can be thought of as all the points that are either in *S* or "near" *S*. A point which is in the closure of *S* is a point of closure of *S*. The notion of closure is in many ways dual to the notion of interior.

In geometry, topology, and related branches of mathematics, a **closed set** is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

In mathematics, specifically in topology, the **interior** of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an **interior point** of S.

In topology, a **discrete space** is a particularly simple example of a topological space or similar structure, one in which the points form a *discontinuous sequence*, meaning they are *isolated* from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.

In topology and mathematics in general, the **boundary** of a subset *S* of a topological space *X* is the set of points which can be approached both from *S* and from the outside of *S*. More precisely, it is the set of points in the closure of *S* not belonging to the interior of *S*. An element of the boundary of *S* is called a **boundary point** of *S*. The term **boundary operation** refers to finding or taking the boundary of a set. Notations used for boundary of a set *S* include bd(*S*), fr(*S*), and . Some authors use the term **frontier** instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, *Metric Spaces* by E. T. Copson uses the term boundary to refer to Hausdorff's **border**, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term **residue**, which is defined as the intersection of a set with the closure of the border of its complement.

In mathematics, a **ball** is the volume space bounded by a sphere; it is also called a **solid sphere**. It may be a **closed ball** or an **open ball**.

In mathematics, a **Baire space** is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.

In mathematics, more specifically in topology, an **open map** is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image is open in Likewise, a **closed map** is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

In topology and related areas of mathematics, a **neighbourhood** is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set.

In mathematics, more specifically in point-set topology, the **derived set** of a subset of a topological space is the set of all limit points of It is usually denoted by

In mathematics, the **support** of a measure *μ* on a measurable topological space is a precise notion of where in the space *X* the measure "lives". It is defined to be the largest (closed) subset of *X* for which every open neighbourhood of every point of the set has positive measure.

In the study of metric spaces in mathematics, there are various notions of two metrics on the same underlying space being "the same", or **equivalent**.

In mathematics, the concept of a **generalised metric** is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field.

In geometric topology, a **cellular decomposition***G* of a manifold *M* is a decomposition of *M* as the disjoint union of cells.

In topology, a subfield of mathematics, *filters* are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called *ultrafilters* have many useful technical properties and they may often be used in place of arbitrary filters.

- ↑ Metric space#Open and closed sets.2C topology and convergence
- ↑ Steen, L. A.; Seebach, J. A. (1995),
*Counterexamples in Topology*, Dover, ISBN 0-486-68735-X

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.