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A topology induced by the metric g defined on a metric space X. The open sets are all subsets that can be realized as the unions of open balls B(x_0,r)={x in X|g(x_0,x)<r}, ...

Also called indiscrete topology, the trivial topology is the smallest topology on a set X, namely the one in which the only open sets are the empty set and the entire set X. ...

Let X=(X,tau) be a topological vector space whose continuous dual X^* may or may not separate points (i.e., may or may not be T2). The weak-* (pronounced "weak star") ...

Algebraic topology is the study of intrinsic qualitative aspects of spatial objects (e.g., surfaces, spheres, tori, circles, knots, links, configuration spaces, etc.) that ...

Let X=(X,tau) be a topological vector space whose continuous dual X^* separates points (i.e., is T2). The weak topology tau_w on X is defined to be the coarsest/weakest ...

A topology that is "potentially" a metric topology, in the sense that one can define a suitable metric that induces it. The word "potentially" here means that although the ...

A topology on a set X whose open sets are the unions of open balls B(X_0,r)={x in x|g(x_0,x)<r}, where g is a pseudometric on X, x_0 is any point of X, and r>0. There is a ...

The norm topology on a normed space X=(X,||·||_X) is the topology tau consisting of all sets which can be written as a (possibly empty) union of sets of the form B_r(x)={y in ...

Noncommutative topology is a recent program having important and deep applications in several branches of mathematics and mathematical physics. Because every commutative ...

A topological space is second countable if it has a countable topological basis.