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The set Γ of all open intervals in forms a basis for the Euclidean topology on .. A non-empty family of subsets of a set X that is closed under finite intersections of two or more sets, which is called a π-system on X, is necessarily a base for a topology on X if and only if it covers X.
The product topology, ... is a basis for the product topology of ... is the topology generated by sets of the form (), where ...
The second subbase generates the usual topology as well, since the open intervals (,) with , rational, are a basis for the usual Euclidean topology. The subbase consisting of all semi-infinite open intervals of the form ( − ∞ , a ) {\displaystyle (-\infty ,a)} alone, where a {\displaystyle a} is a real number, does not generate the usual ...
The Golomb topology is connected, [6] [2] [13] but not locally connected. [6] [13] [14] The Kirch topology is both connected and locally connected. [9] [3] [13] The integers with the Furstenberg topology form a homogeneous space, because it is a topological ring — in some sense, the only topology on for which it is a ring. [15]
The standard topology on R is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set.
The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set and the whole space as open sets. In function spaces and spaces of measures there are often a number of possible topologies.
The basis for a free group is not uniquely determined. Being characterized by a universal property is the standard feature of free objects in universal algebra . In the language of category theory , the construction of the free group (similar to most constructions of free objects) is a functor from the category of sets to the category of groups .
Though the subspace topology of Y = {−1} ∪ {1/n } n∈N in the section above is shown not to be generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the discrete topology on Y (the topology in which every subset of Y is open ...