### What does **open set** mean?

# Definitions for open set

open set

#### This dictionary definitions page includes all the possible meanings, example usage and translations of the word **open set**.

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### Wiktionary

open setnoun

Informally, a set such that the target point of a movement by a small amount in any direction from any point in the set is still in the set; exemplified by a full circle without its boundary.

open setnoun

A set which can be described as an (arbitrary) union of open balls. Equivalently, a set such that for every point in it, there is an open ball centered at that point, such that that open ball is contained by the set.

open setnoun

Most generally, a member of the topology of a given topological space.

### Wikipedia

Open set

In mathematics, an open set is a generalization of open interval in the real line. In a metric space (a set along with a distance defined between any two points), an open set is a set that, along with every point P, contains all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P). More generally, an open set is a member of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, every subset can be open (the discrete topology), or no subset can be open except the space itself and the empty set (the indiscrete topology). In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such as continuity, connectedness, and compactness, which were originally defined by means of a distance. The most common case of a topology without any distance is given by manifolds, which are topological spaces that, near each point, resemble an open set of a Euclidean space, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology, which is fundamental in algebraic geometry and scheme theory.

### Freebase

Open set

In topology, a set U is called an open set if it does not contain any of its boundary points. When dealing with metric spaces, there is a well-defined distance between any two points. A subset U of a metric space is open if, for every point p in U, there is some positive distance such that every point which is at least this close to p is also contained in U. The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Concepts that use notions of nearness, such as the continuity of functions, can be translated into the language of open sets. In point-set topology, open sets are used to distinguish between points and subsets of a space. The degree to which any two points can be separated is specified by the separation axioms. The collection of all open sets in a space defines the topology of the space. Functions from one topological space to another that preserve the topology are the continuous functions. Although open sets and the topologies that they comprise are of central importance in point-set topology, they are also used as an organizational tool in other important branches of mathematics. Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.

### Anagrams for open set »

openest

pentose

### Numerology

Chaldean Numerology

The numerical value of open set in Chaldean Numerology is:

**1**Pythagorean Numerology

The numerical value of open set in Pythagorean Numerology is:

**4**

## Translations for **open set**

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"open set." *Definitions.net.* STANDS4 LLC, 2023. Web. 2 Dec. 2023. <https://www.definitions.net/definition/open+set>.

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