Scattered space

Last updated

In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset. [1] [2] Equivalently, every nonempty subset A of X contains a point isolated in A.

Contents

A subset of a topological space is called a scattered set if it is a scattered space with the subspace topology.

Examples

Properties

Notes

  1. Steen & Seebach, p. 33
  2. Engelking, p. 59
  3. See proposition 2.8 in Al-Hajri, Monerah; Belaid, Karim; Belaid, Lamia Jaafar (2016). "Scattered Spaces, Compactifications and an Application to Image Classification Problem". Tatra Mountains Mathematical Publications. 66: 1–12. doi: 10.1515/tmmp-2016-0015 . S2CID   199470332.
  4. "General topology - in a $T_0$ space the union of two scattered sets is scattered".
  5. "General topology - Second countable scattered spaces are countable".
  6. Willard, problem 30E, p. 219
  7. "General topology - Uniqueness of decomposition into perfect set and scattered set".
  8. "Real analysis - is Cantor-Bendixson theorem right for a general second countable space?".

Related Research Articles

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 topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms.

<span class="mw-page-title-main">General topology</span> Branch 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.

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

In the mathematical field of general topology, a meagre set is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms.

In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated from the German nouns Gebiet'open set' and Durchschnitt'intersection'. Historically Gδ sets were also called inner limiting sets, but that terminology is not in use anymore. Gδ sets, and their dual, F𝜎 sets, are the second level of the Borel hierarchy.

In mathematics, a cofinite subset of a set is a subset whose complement in is a finite set. In other words, contains all but finitely many elements of If the complement is not finite, but is countable, then one says the set is cocountable.

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 general topology and related areas of mathematics, the disjoint union of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other.

In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of It is usually denoted by

In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set is perfect if , where denotes the set of all limit points of , also known as the derived set of .

In mathematics, the particular point topology is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and pX. The collection

In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.

In general topology, a subset of a topological space is said to be dense-in-itself or crowded if has no isolated point. Equivalently, is dense-in-itself if every point of is a limit point of . Thus is dense-in-itself if and only if , where is the derived set of .

In mathematics, a topological space is called collectionwise normal if for every discrete family Fi (iI) of closed subsets of there exists a pairwise disjoint family of open sets Ui (iI), such that FiUi. Here a family of subsets of is called discrete when every point of has a neighbourhood that intersects at most one of the sets from . An equivalent definition of collectionwise normal demands that the above Ui (iI) themselves form a discrete family, which is stronger than pairwise disjoint.

In mathematics, a topological space is said to be limit point compact or weakly countably compact if every infinite subset of has a limit point in This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.

In mathematics, particularly topology, a Gδ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.

In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it. Formally, is dense in if the smallest closed subset of containing is itself.

In mathematics, more specifically general topology, the divisor topology is a specific topology on the set of positive integers greater than or equal to two. The divisor topology is the poset topology for the partial order relation of divisibility of integers on .

In general topology, a branch of mathematics, the Appert topology, named for Antoine Appert (1934), is a topology on the set X = {1, 2, 3, ...} of positive integers. In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer. The space X with the Appert topology is called the Appert space.

References