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 this closed version, as they prefer to use the term 'closed topologist's sine curve' to refer to another curve. [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 ( HOWS-dorf, HOWZ-dorf), separated space or T2 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" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and 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 or form, introduced to mathematics by Henri Poincaré in 1895.
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.
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 that is also a topological space with the property that the vector space operations are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space. One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs.
In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I. In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology.
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 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 laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.
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 that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed 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.
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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.
