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 (possibly very small) 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 mani

Other articles related to "open set, set":

Euler Sequence - Geometric Interpretation
... A function (defined on some open set) on gives rise by pull-back to a 0-homogeneous function on V (again partially defined) ... Recall that a vector field on an open set U of the projective space can be defined as a derivation of the functions defined on this open set ...
Beauville–Laszlo Theorem - The Theorem
... is its complement Df = Spec Af, the principal open set determined by f, while  is an "infinitesimal neighborhood" D = Spec  of (f) ... we can restrict it to both the principal open set Df and the infinitesimal neighborhood Spec Â, yielding an Af-module F and an Â-module G ... Geometrically, we are given a scheme X and both an open set Df and a "small" neighborhood D of its closed complement (f) on Df and D we are given two sheaves which agree on the ...
Topological Indistinguishability - Properties - Equivalent Conditions
... Then the following statements are equivalent x ≡ y for each open set U in X, either U contains both x and y or neither of them Nx = Ny x ∈ cl{y} and y ∈ cl{x} cl{x} = cl{y} x ∈ ∩Ny and ... in particular, for regular spaces), the following statements are equivalent x ≡ y for each open set U, if x ∈ U then y ∈ U Nx ⊂ Ny x ∈ cl{y} x ∈ ∩Ny x belongs to every closed set ...
Adherent Point
... of a subset A of a topological space X, is a point x in X such that every open set containing x contains at least one point of ... in that for a limit point it is required that every open set containing contains at least one point of A different from x ... Intuitively, having an open set A defined as the area within (but not including) some boundary, the adherent points of A are those of A including the boundary ...

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