In mathematics, a **closed manifold** is a manifold without boundary that is compact.

In comparison, an **open manifold** is a manifold without boundary that has only *non-compact* components.

The only connected one-dimensional example is a circle. The torus and the Klein bottle are closed. A line is not closed because it is not compact. A closed disk is compact, but is not a closed manifold because it has a boundary.

For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.

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Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical conditions),thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesn’t cover some basic objects such as a closed disk, so authors sometimes define a manifold with boundary and abusively say *manifold* without reference to the boundary. But normally, a **compact manifold** (compact with respect to its underlying topology) can synonymously be used for **closed manifold** if the usual definition for manifold is used.

The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and a manifold, but not a closed manifold.

The notion of a "closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive Ricci curvature.

In topology and related branches of mathematics, a **connected space** is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.

In physics and mathematics, the **dimension** of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.

In the part of mathematics referred to as topology, a **surface** is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

In mathematics, **topology** is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

In topology and mathematics in general, the **boundary** of a subset *S* of a topological space *X* is the set of points which can be approached both from *S* and from the outside of *S*. More precisely, it is the set of points in the closure of *S* not belonging to the interior of *S*. An element of the boundary of *S* is called a **boundary point** of *S*. The term **boundary operation** refers to finding or taking the boundary of a set. Notations used for boundary of a set *S* include bd(*S*), fr(*S*), and . Some authors use the term **frontier** instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, *Metric Spaces* by E. T. Copson uses the term boundary to refer to Hausdorff's **border**, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term **residue**, which is defined as the intersection of a set with the closure of the border of its complement.

The **shape of the universe**, in physical cosmology, is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes general global properties of its shape as of a continuous object. The spatial curvature is related to general relativity, which describes how spacetime is curved and bent by mass and energy, while the spatial topology cannot be determined from its curvature; locally indistinguishable spaces with different topologies exist mathematically.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In topology, the **long line** is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore, it serves as one of the basic counterexamples of topology. Intuitively, the usual real-number line consists of a countable number of line segments [0, 1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.

In mathematics, a **foliation** is an equivalence relation on an *n*-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension *p*, modeled on the decomposition of the real coordinate space **R**^{n} into the cosets *x* + **R**^{p} of the standardly embedded subspace **R**^{p}. The equivalence classes are called the **leaves** of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class *C ^{r}* it is usually understood that

In mathematics, **cobordism** is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds of the same dimension are *cobordant* if their disjoint union is the *boundary* of a compact manifold one dimension higher.

In topology, a topological space is called **simply connected** if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.

In mathematics, a **3-manifold** is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In mathematical knot theory, a **link** is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a *trivial* reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

In mathematics, a **pair of pants** is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants.

In topology, a branch of mathematics, the **ends** of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the **end compactification**.

In mathematical physics, **global hyperbolicity** is a certain condition on the causal structure of a spacetime manifold. It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is

In mathematics, specifically geometry and topology, the **classification of manifolds** is a basic question, about which much is known, and many open questions remain.

In mathematics, a **Klein surface** is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves over the complex numbers analytically, the former are used to study algebraic curves over the real numbers analytically. Klein surfaces were introduced by Felix Klein in 1882.

- Michael Spivak:
*A Comprehensive Introduction to Differential Geometry.*Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, ISBN 0-914098-70-5.

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