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In mathematics, a base (or basis; pl.: 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 .
The product topology on X is the topology generated by sets of the form p i −1 (U), where i is in I and U is an open subset of X i. In other words, the sets {p i −1 (U)} form a subbase for the topology on X. A subset of X is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form p i −1 (U).
Then τ is called a topology on X if: Both the empty set and X are elements of τ. Any union of elements of τ is an element of τ. Any intersection of finitely many elements of τ is an element of τ. If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation X τ may be used to denote a set X endowed with the ...
The left order topology on X is the topology having as a base all intervals of the form (,) = {<}, together with the set X. The left and right order topologies can be used to give counterexamples in general topology.
The weight of a space X is the smallest cardinal number κ such that X has a base of cardinal κ. (Note that such a cardinal number exists, because the entire topology forms a base, and because the class of cardinal numbers is well-ordered.) Well-connected See Ultra-connected. (Some authors use this term strictly for ultra-connected compact ...
An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser. One can also compare topologies using neighborhood bases. Let τ 1 and τ 2 be two topologies on a set X and let B i (x) be a local base for the topology τ i at x ∈ X for i = 1,2.
Suppose that X is a regular space. Then, given any point x and neighbourhood G of x, there is a closed neighbourhood E of x that is a subset of G. In fancier terms, the closed neighbourhoods of x form a local base at x. In fact, this property characterises regular spaces; if the closed neighbourhoods of each point in a topological space form a ...
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.More explicitly, a topological space is second-countable if there exists some countable collection = {} = of open subsets of such that any open subset of can be written as a union of elements of some subfamily of .