English Dictionary

topological space

名詞

フォーマル 広義

(topology, formally) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);

日本語の意味

位相空間：集合 X と、その上で定められる位相 τ（X の部分集合の集まりで、特定の公理―例えば、X と空集合を含み、任意の合併及び有限個の交わりが位相に含まれる―を満たす）が定める順序対のこと。

To prove that a function is continuous, it suffices to show that the preimage of every open set in the codomain is open in the topological space.

関数が連続であることを証明するには、像の任意の開集合の逆像が位相空間（X は集合で τ は位相である順序対 (X, τ)）において開集合であることを示せば十分である。

意味（1）

(topology, formally) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);

In order to obtain intuitive insight into special classes of topological spaces we can proceed in several ways, only a few of which will be pursued in this chapter. For instance, we can seek to describe important topological spaces by means of enough of their properties to completely characterize them, up to homeomorphism.

意味（2）

(loosely) the set X.

topological spaces （ plural ）