Concept
Product topology
Related concepts (25)
Metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric such that the topology induced by is Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable.
Complete metric space
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below).
Topology
In mathematics, topology (from the Greek words τόπος, and λόγος) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space is a if it is a connected space when viewed as a subspace of . Some related but stronger conditions are path connected, simply connected, and -connected.
Quotient space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its under the canonical projection map is open in the original topological space.
Base (topology)
In mathematics, a base (or basis; : bases) for the topology τ of a topological space (X, τ) is a family of open subsets of X such that every open set of the topology is equal to the union of some sub-family of . For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology.
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed Tychonoff), who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article of Tychonoff, A.
Subbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space with topology is a subcollection of that generates in the sense that is the smallest topology containing as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Suppose that is a set and is a topological space, such as the real or complex numbers or a metric space, for example. A net or sequence of functions all having the same domain and codomain is said to converge pointwise to a given function often written as if (and only if) The function is said to be the pointwise limit function of the Sometimes, authors use the term bounded pointwise convergence when there is a constant such that .
Initial topology
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set with respect to a family of functions on is the coarsest topology on that makes those functions continuous. The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.