In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A neighbourhood or neighborhood (see spelling differences) is a geographically localised Community within a larger City, Town or
Properties of a single space
A topological space is sometimes said to exhibit a property locally if the property is exhibited "near" each point in one of the following different senses:
- Each point has a neighborhood exhibiting the property;
- Each point has a neighborhood base of sets exhibiting the property. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the
Sense (2) is in general stronger than sense (1), and caution must be taken to distinguish between the two senses. For example, some variation in the definition of locally compact arises from different senses of the term locally. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks
Examples
- Locally compact topological spaces
- Locally connected and Locally path-connected topological spaces
Properties of a pair of spaces
Given some notion of equivalence (e. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks In Topology and other branches of Mathematics, a Topological space is said to be locally connected at x (where x is a point In Topology and other branches of Mathematics, a Topological space is said to be locally connected at x (where x is a point g. , homeomorphism, diffeomorphism, isometry) between topological spaces, two spaces are locally equivalent if every point of the first space has a neighborhood which is equivalent to a neighborhood of the second space. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable For the Mechanical engineering and Architecture usage see Isometric projection. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.
For instance, the circle and the line are very different objects. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the One cannot stretch the circle to look like the line, nor compress the line to fit on the circle without gaps or overlaps. However, a small piece of the circle can be stretched and flattened out to look like a small piece of the line. For this reason, one may say that the circle and the line are locally equivalent.
Similarly, the sphere and the plane are locally equivalent. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe A small enough observer standing on the surface of a sphere (e. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. g. , a person and the Earth) would find it indistinguishable from a plane.
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