Manifold

Topological manifolds

Topological manifold without boundary

The prototypical example of a topological manifold without boundary is Euclidean space. A general manifold without boundary looks locally, as a topological space, like Euclidean space. This is formalized by requiring that a manifold without boundary is a non-empty topological space in which every point has an open neighbourhood homeomorphic to (an open subset of) Rn (Euclidean n-space). Another way of saying this, using charts, is that a manifold without boundary is a non-empty topological space in which at every point there is an Rn-chart.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Topological manifold with boundary

More generally it is possible to allow a topological manifold to have a boundary. The prototypical example of a topological manifold with boundary is the Euclidean closed half-space. Most points in Euclidean closed half-space, those not on the boundary, have a neighbourhood homeomorphic to Euclidean space in addition to having a neighbourhood homeomorphic to Euclidean closed half-space, but the points on the boundary only have neighbourhoods homeomorphic to Euclidean closed half-space and not to Euclidean space. Thus we need to allow for two kinds of points in our topological manifold with boundary: points in the interior and points in the boundary. Points in the interior will, as before, have neighbourhoods homeomorphic to Euclidean space, but may also have neighbourhoods homeomorphic to Euclidean closed half-space. Points in the boundary will have neighbourhoods homeomorphic to Euclidean closed half-space. Thus a topological manifold with boundary is a non-empty topological space in which at each point there is an Rn-chart or an [0,∞)×Rn−1-chart. The set of points at which there are only [0,∞)×Rn−1-charts is called the boundary and its complement is called the interior. The interior is always non-empty and is a topological n-manifold without boundary. If the boundary is non-empty then it is a topological (n-1)-manifold without boundary. If the boundary is empty, then we regain the definition of a topological manifold without boundary.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Examples

The open interval (0,1) is a one-dimensional manifold without boundary. The closed interval is a one-dimensional manifold with boundary. Every connected one-dimensional manifold is homeomorphic to one or the other of these. The closed unit disk is a two-dimensional manifold with boundary. The plane R2 and the sphere S2 are two-dimensional manifolds without boundary. They are not homeomorphic, since the former is non-compact and the latter compact. The torus T2 and the projective plane P2 are other examples of compact two-dimensional manifolds without boundary. The projective plane is an example of a non-orientable manifold. Every compact, connected two-manifold is homeomorphic to a sphere, to a torus, to a connected sum of torii, or to a connected sum of torii and one projective plane. This is the solution to the classification problem for compact, connected two-manifolds. In higher dimensions, the classification problem has not yet been solved, and is an active area of mathematical research. Higher dimensional manifolds are harder to visualize, but are important in mathematics and physics. Space-time may be a four-dimensional manifold, or may have singular points (singularities) at the big bang and at black holes where no manifold structure exists. Infinite dimensional manifolds also exist, at least mathematically.

Related Topics:
Open interval - Closed interval - Closed unit disk - Plane - Sphere - Compact - Projective plane - Non-orientable - Classification problem - Singular points - Big bang - Black holes

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Properties

A manifold with empty boundary is said to be closed if it is compact, and open if it is not compact. See closed manifold.

Related Topics:
Compact - Closed manifold

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Manifolds inherit many of the local properties of Euclidean space.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In particular, they are locally path-connected, locally compact and locally metrizable.

Related Topics:
Locally path-connected - Locally compact - Metrizable

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Being locally compact Hausdorff spaces they are necessarily Tychonoff spaces. Requiring a manifold to be Hausdorff may seem strange; it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the real line with two origins is not Hausdorff, because the two origins cannot be separated.

Related Topics:
Tychonoff space - Homeomorphic - Counterexample - Real line

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A topological space is said to be homogeneous if its homeomorphism group acts transitively on it. Every connected manifold without boundary is homogeneous, but manifolds with nonempty boundary are not homogeneous.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It can be shown that a manifold is metrizable if and only if it is paracompact.

Related Topics:
Metrizable - Paracompact

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Non-paracompact manifolds (such as the long line) are generally regarded as pathological, so it's common to add paracompactness to the definition of an n-manifold.

Related Topics:
Long line - Pathological

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Sometimes n-manifolds are defined to be second-countable, which is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Note that every compact manifold is second-countable, and every second-countable manifold is paracompact.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~