Closed set in the context of "Limit point"

In mathematics, the support of a real-valued function ${\displaystyle f}$ is the subset of the function's domain consisting of those elements that are not mapped to zero. If the domain of ${\displaystyle f}$ is a topological space, then the support of ${\displaystyle f}$ is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis.

Closeness is a basic concept in topology and related areas in mathematics. Intuitively, we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in a metric space where a notion of distance between elements of the space is defined, but it can be generalized to topological spaces where we have no concrete way to measure distances.

The closure operator closes a given set by mapping it to a closed set which contains the original set and all points close to it. The concept of closeness is related to limit point.

In mathematics, an open set is a generalization of an open interval in the real line.

In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.

Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility.

In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to ${\displaystyle {\sqrt {n}}}$.

An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that every infinite sequence of points has limiting values. For example, the real line is not compact since the sequence of natural numbers has no real limiting value. The open interval (0,1) is not compact because it excludes the limiting values 0 and 1, whereas the closed interval [0,1] is compact. Similarly, the space of rational numbers ${\displaystyle \mathbb {Q} }$ is not compact, because every irrational number is the limit of the rational numbers that are lower than it. On the other hand, the extended real number line is compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.

One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence ⁠1/2⁠, ⁠4/5⁠, ⁠1/3⁠, ⁠5/6⁠, ⁠1/4⁠, ⁠6/7⁠, ... accumulate to 0 (while others accumulate to 1). Since neither 0 nor 1 are members of the open unit interval (0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering ${\displaystyle \mathbb {R} ^{1}}$ (the real number line), the sequence of points 0,  1,  2,  3, ... has no subsequence that converges to any real number.

In mathematics, specifically in functional analysis, a C-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:

• A is a topologically closed set in the norm topology of operators.
• A is closed under the operation of taking adjoints of operators.

Another important class of non-Hilbert C*-algebras includes the algebra ${\displaystyle C_{0}(X)}$ of complex-valued continuous functions on X that vanish at infinity, where X is a locally compact Hausdorff space.