In mathematics, a **cross-cap** is a two-dimensional surface in 3-space that is one-sided and the continuous image of a Möbius strip that intersects itself in an interval. In the domain, the inverse image of this interval is a longer interval that the mapping into 3-space "folds in half". At the point where the longer interval is folded in half in the image, the nearby configuration is that of the Whitney umbrella.

The interval of self intersection precludes the cross-cap from being homeomorphic to the Möbius strip, but there are only two points in the image (the endpoints of the interval of self-intersection) where the image cannot be that of an immersion. The bounding edge of a cross-cap is a simple closed loop. Like certain versions of the Möbius strip, it may take the form of a symmetrical circle.

A cross-cap that has been closed up by gluing a disc to its boundary is a model of the real projective plane P^{2} (again with an interval of self-intersection, and two points where this model is not an immersion of P^{2}).

Two cross-caps glued together at their boundaries form a model of the Klein bottle, this time with two intervals of self-intersection and four points where this model is not an immersion.

An important theorem of topology, the classification theorem for surfaces, states that each two-dimensional compact manifold without boundary is homeomorphic to a sphere with some number (possibly 0) of "handles" and 0, 1, or 2 cross-caps.

In topology, a branch of mathematics, the **Klein bottle** is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.

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 related branches of mathematics, a **topological space** may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

In mathematics, a **Möbius strip**, **band**, or **loop**, also spelled **Mobius** or **Moebius**, is a surface with only one side and only one boundary curve. The Möbius strip is the simplest non-orientable surface. It can be realized as a ruled surface. Its discovery is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858, though similar structures can be seen in Roman mosaics c. 200–250 AD. Möbius published his results in his articles "Theorie der elementaren Verwandtschaft" (1863) and "Ueber die Bestimmung des Inhaltes eines Polyëders" (1865).

In geometry, **Boy's surface** is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane *could not* be immersed in 3-space.

In geometry, a **torus** is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

In mathematics, **orientability** is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of a normal vector allows one to use the right-hand rule to define a "clockwise" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a "clockwise" orientation for all loops in the manifold. Equivalently, a surface is **orientable** if a two-dimensional figure in the space cannot be moved continuously on that surface and back to its starting point so that it looks like its own mirror image.

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 mathematical analysis, a **space-filling curve** is a curve whose range contains the entire 2-dimensional unit square. Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called *Peano curves*, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.

In mathematics, the **real projective plane** is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in **R**^{3} passing through the origin.

In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:

In mathematics, **low-dimensional topology** is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

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 mathematics, a **solid Klein bottle** is a three-dimensional topological space whose boundary is the Klein bottle.

In mathematics, an **immersion** is a differentiable function between differentiable manifolds whose derivative is everywhere injective. Explicitly, *f* : *M* → *N* is an immersion if

In differential geometry, Mikhail Gromov's **filling area conjecture** asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.

The **Alexandrov uniqueness theorem** is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, and it characterizes the metric spaces that come from the surface distances on polyhedra. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s.

In geometry, the **bundle theorem** is in the simplest case a statement on six circles and eight points in the real Euclidean plane. In general it is a property of a Möbius plane that is fulfilled by *ovoidal* Möbius planes only.

In mathematics, a **planar Riemann surface** is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910 as a generalization of the uniformization theorem that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed.

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