In the branch of mathematics known as topology, the **topologist's sine curve** or **Warsaw sine curve** is a topological space with several interesting properties that make it an important textbook example.

It can be defined as the graph of the function sin(1/*x*) on the half-open interval (0, 1], together with the origin, under the topology induced from the Euclidean plane:

The topologist's sine curve *T* is connected but neither locally connected nor path connected. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a path.

The space *T* is the continuous image of a locally compact space (namely, let *V* be the space {−1} ∪ (0, 1], and use the map *f* from *V* to *T* defined by *f*(−1) = (0,0) and *f*(*x*) = (*x*, sin(1/*x*)) for *x* > 0), but *T* is not locally compact itself.

The topological dimension of *T* is 1.

Two variants of the topologist's sine curve have other interesting properties.

The **closed topologist's sine curve** can be defined by taking the topologist's sine curve and adding its set of limit points, ; some texts define the topologist's sine curve itself as just this closed version.^{ [1] } This space is closed and bounded and so compact by the Heine–Borel theorem, but has similar properties to the topologist's sine curve—it too is connected but neither locally connected nor path-connected.

The **extended topologist's sine curve** can be defined by taking the closed topologist's sine curve and adding to it the set . It is arc connected but not locally connected.

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 topology and related branches of mathematics, a **Hausdorff space**, **separated space** or **T _{2} space** is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T

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.

In topology and related areas of mathematics, a **product space** is the Cartesian product of a family of topological spaces equipped with a natural topology called the **product topology**. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

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 vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

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 and related areas of mathematics, the **quotient space** of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the **quotient topology**, that is, with the finest topology that makes continuous the canonical projection map. In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.

In topology and related branches of mathematics, a **totally disconnected space** is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every topological space, the singletons are connected; in a totally disconnected space, these are the *only* connected proper subsets.

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

* Counterexamples in Topology* is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.

In mathematics, a **path** in a topological space is a continuous function from the closed unit interval into

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 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. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.

The **Vietoris–Begle mapping theorem** is a result in the mathematical field of algebraic topology. It is named for Leopold Vietoris and Edward G. Begle. The statement of the theorem, below, is as formulated by Stephen Smale.

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, particularly topology, a **comb space** is a particular subspace of that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The **deleted comb space** is a variation on the comb space.

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.

- ↑ Munkres, James R (1979).
*Topology; a First Course*. Englewood Cliffs. p. 158. ISBN 9780139254956.

- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978],
*Counterexamples in Topology*(Dover reprint of 1978 ed.), Mineola, NY: Dover Publications, Inc., pp. 137–138, ISBN 978-0-486-68735-3, MR 1382863 - Weisstein, Eric W. "Topologist's Sine Curve".
*MathWorld*.

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