In mathematics, a topological space *X* is **contractible** if the identity map on *X* is null-homotopic, i.e. if it is homotopic to some constant map.^{ [1] }^{ [2] } Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.

A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial.

For a topological space *X* the following are all equivalent:

*X*is contractible (i.e. the identity map is null-homotopic).*X*is homotopy equivalent to a one-point space.*X*deformation retracts onto a point. (However, there exist contractible spaces which do not*strongly*deformation retract to a point.)- For any space
*Y*, any two maps*f*,*g*:*Y*→*X*are homotopic. - For any space
*Y*, any map*f*:*Y*→*X*is null-homotopic.

The cone on a space *X* is always contractible. Therefore any space can be embedded in a contractible one (which also illustrates that subspaces of contractible spaces need not be contractible).

Furthermore, *X* is contractible if and only if there exists a retraction from the cone of *X* to *X*.

Every contractible space is path connected and simply connected. Moreover, since all the higher homotopy groups vanish, every contractible space is *n*-connected for all *n* ≥ 0.

A topological space *X* is **locally contractible at a point***x* if for every neighborhood *U* of *x* there is a neighborhood *V* of *x* contained in *U* such that the inclusion of *V* is nulhomotopic in *U*. A space is **locally contractible** if it's locally contractible at every point. This definition is occasionally referred to as the "geometric topologist's locally contractible," though is the most common usage of the term. In Hatcher's standard Algebraic Topology text, this definition is referred to as "weakly locally contractible," though that term has other uses.

If every point has a local base of contractible neighborhoods, then we say that *X* is **strongly locally contractible**. Contractible spaces are not necessarily locally contractible nor vice versa. For example, the comb space is contractible but not locally contractible (if it were, it would be locally connected which it is not). Locally contractible spaces are locally *n*-connected for all *n* ≥ 0. In particular, they are locally simply connected, locally path connected, and locally connected. The circle is (strongly) locally contractible but not contractible.

Strong local contractibility is a strictly stronger property than local contractibility; the counterexamples are sophisticated, the first being given by Borsuk and Mazurkiewicz in their paper *Sur les rétractes absolus indécomposables*, C.R.. Acad. Sci. Paris 199 (1934), 110-112).

There is some disagreement about which definition is the "standard" definition of local contractibility; the first definition is more commonly used in geometric topology, especially historically, whereas the second definition fits better with the typical usage of the term "local" with respect to topological properties. Care should always be taken regarding the definitions when interpreting results about these properties.

- Any Euclidean space is contractible, as is any star domain on a Euclidean space.
- The Whitehead manifold is contractible.
- Spheres of any finite dimension are not contractible.
- The unit sphere in an infinite-dimensional Hilbert space is contractible.
- The house with two rooms is standard example of a space which is contractible, but not intuitively so.
- The Dunce hat is contractible, but not collapsible.
- The cone on a Hawaiian earring is contractible (since it is a cone), but not locally contractible or even locally simply connected.
- All manifolds and CW complexes are
*locally*contractible, but in general not contractible. - The Warsaw circle is obtained by "closing up" the topologist's sine curve by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose homotopy groups are all trivial, but it is not contractible.

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 the mathematical field of algebraic topology, the **fundamental group** of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups.

In the mathematical field of topology, a **homeomorphism**, **topological isomorphism**, or **bicontinuous function** is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called **homeomorphic**, and from a topological viewpoint they are the same. The word *homeomorphism* comes from the Greek words *ὅμοιος* (*homoios*) = similar or same and *μορφή* (*morphē*) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, a **topological group** is a group G together with a topology on G such that both the group's binary operation and the function mapping group elements to their respective inverses are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

In mathematics, the **real line**, or **real number line** is the line whose points are the real numbers. That is, the real line is the set **R** of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

In mathematics, a **chain complex** is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In topology, a branch of mathematics, two continuous functions from one topological space to another are called **homotopic** if one can be "continuously deformed" into the other, such a deformation being called a **homotopy** between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

In mathematics, specifically algebraic topology, a **covering map** is a continuous function from a topological space to a topological space such that each point in has an open neighbourhood **evenly covered** by . In this case, is called a **covering space** and the **base space** of the covering projection. The definition implies that every covering map is a local homeomorphism.

A **CW complex** is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. The *C* stands for "closure-finite", and the *W* for "weak" topology. A CW complex can be defined inductively.

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 topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In topology, a branch of mathematics, a **retraction** is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a **retract** of the original space. A **deformation retraction** is a mapping that captures the idea of *continuously shrinking* a space into a subspace.

In the mathematical branch of algebraic topology, specifically homotopy theory, ** n-connectedness** generalizes the concepts of path-connectedness and simple connectedness. To say that a space is

In topology and other branches of mathematics, a topological space *X* is **locally connected** if every point admits a neighbourhood basis consisting entirely of open, connected sets.

In mathematics, an **acyclic space** is a topological space *X* in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of *X* are isomorphic to the corresponding homology groups of a point.

In mathematics, especially in the area of topology known as algebraic topology, an **induced homomorphism** is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space *X* to a space *Y* induces a group homomorphism from the fundamental group of *X* to the fundamental group of *Y*.

In the mathematical field of point-set topology, a **continuum** is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. **Continuum theory** is the branch of topology devoted to the study of continua.

In mathematics, a **weak equivalence** is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category.

- ↑ Munkres, James R. (2000).
*Topology*(2nd ed.). Prentice Hall. ISBN 0-13-181629-2. - ↑ Hatcher, Allen (2002).
*Algebraic Topology*. Cambridge University Press. ISBN 0-521-79540-0.

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