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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.
In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: (0,1], [0,1), and (0,1). However, the notation I is most commonly reserved for the closed interval [0,1].
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. Then τ 1 ⊆ τ 2 if and only if for all x ∈ X, each open set U 1 in B 1 (x) contains some open set U 2 in B 2 (x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.
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 ...
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.
In the following, represents the real numbers with their usual topology. The subspace topology of the natural numbers, as a subspace of , is the discrete topology.; The rational numbers considered as a subspace of do not have the discrete topology ({0} for example is not an open set in because there is no open subset of whose intersection with can result in only the singleton {0}).
C 0 is isomorphic to Z 3 with basis (v 0), (v 1), (v 2), C 1 is isomorphic to Z 3 with a basis given by the oriented 1-simplices (v 0, v 1), (v 0, v 2), and (v 1, v 2). C 2 is the trivial group, since there is no simplex like (,,) because the triangle has been supposed without its interior. So are the chain groups in other dimensions.
In any metric space, the open balls form a base for a topology on that space. [1] The Euclidean topology on R n {\displaystyle \mathbb {R} ^{n}} is the topology generated by these balls.