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However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of closed sets, of closure and interior operators, and of convergence of various types of objects. Each of these can instead be taken as the primary class of objects ...
The foundation of this science, for a space of any dimension, was created by Henri Poincaré. His first article on this topic appeared in 1894. [5] In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane.
For example, if two topological objects have different homotopy groups, they cannot have the same topological structure—a fact that may be difficult to prove using only topological means. For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology ...
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 d is . Metrization theorems are theorems that give sufficient conditions for a topological space to ...
Consider a topological space X and fix a base point of X, then (,) is the fundamental group of the topological space X and the base point , and as a set it has the structure of group; if then let the base point runs over all points of X, and take the union of all (,), then the set we get has only the structure of groupoid (which is called as ...
It's not enough for elements of a topological space to be distinct (that is, unequal); we may want them to be topologically distinguishable. Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them to be separated (in any of various ways). The separation axioms all say, in one way or another, that points or ...
The homotopy hypothesis asks whether a space is something fundamentally algebraic. If one prefers to work with a space instead of a pointed space, there is the notion of a fundamental groupoid (and higher variants): by definition, the fundamental groupoid of a space X is the category where the objects are the points of X and the morphisms are ...
A pointed space means a pair (X,x) with X a topological space and x a point in X, called the base point. The category Top * of pointed spaces has objects the pointed spaces, and a morphism f : X → Y is a continuous map that takes the base point of X to the base point of Y. The naive homotopy category of pointed spaces has the same objects ...