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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.
If is a real or complex vector space and if is the set of all seminorms on then the locally convex TVS topology, denoted by , that induces on is called the finest locally convex topology on . [37] This topology may also be described as the TVS-topology on having as a neighborhood base at the origin the set of all absorbing disks in . [37] Any ...
Filters in topology – Use of filters to describe and characterize all basic topological notions and results. Locally convex topological vector space – Vector space with a topology defined by convex open sets; Neighbourhood (mathematics) – Open set containing a given point; Subbase – Collection of subsets that generate a topology
The structure theorem is of central importance to TDA; as commented by G. Carlsson, "what makes homology useful as a discriminator between topological spaces is the fact that there is a classification theorem for finitely generated abelian groups". [3] (see the fundamental theorem of finitely generated abelian groups).
Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set ℘ and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other.
However, the weak topology can be finer than the product topology, for example if neither X nor Y is locally compact. In this unfavorable case, the product X × Y in the product topology is not a CW complex. On the other hand, the product of X and Y in the category of compactly generated spaces agrees with the weak topology and therefore ...
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.
Firstly, the basic Smith form algorithm has cubic complexity in the size of the matrix involved since it uses row and column operations which makes it unsuitable for large cell complexes. Secondly, the intermediate matrices which result from the application of the Smith form algorithm get filled-in even if one starts and ends with sparse matrices.