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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 ...

The topology on the Cartesian product X×Y of two topological spaces whose open sets are the unions of subsets A×B, where A and B are open subsets of X and Y, respectively. ...

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 ...

An unlabeled transitive digraph with n nodes is called a "topology." The numbers of distinct topologies on n=1, 2, ... nodes are 1, 3, 9, 33, 139, 718, 4535, ... (OEIS ...

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