Search results
Results From The WOW.Com Content Network
The covering dimension of a topological space X is defined to be the minimum value of n such that every finite open cover of X has an open refinement with order n + 1. The refinement B {\displaystyle {\mathfrak {B}}} can always be chosen to be finite. [ 5 ]
The foundation of this science, for a space of any dimension, was created by Henri Poincaré. His first article on this topic appeared in 1894. [5] In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane.
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. [1] A graphical illustration of a zero-dimensional space is a point. [2]
The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a metric space). The other is the Lebesgue covering dimension. The term "topological dimension" is ordinarily understood to ...
The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that, in the case of metric spaces, (n + 1)-dimensional balls have n-dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets. Moreover ...
Door space. A topological space is a door space ... These are precisely the spaces with a small inductive dimension of 0. Almost discrete. A space is almost discrete ...
If a space is compact, then so are all its quotient spaces. A quotient space of a locally compact space need not be locally compact. Dimension. The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
In contrast to compact projective planes there are no topological Möbius planes with circles of dimension >, in particular no compact Möbius planes with a -dimensional point space. [59] All 2-dimensional Möbius planes such that dim Σ ≥ 3 {\displaystyle \dim \Sigma \geq 3} can be described explicitly.