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*.

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

- Every discrete space is scattered.
- Every ordinal number with the order topology is scattered. Indeed, every nonempty subset
*A*contains a minimum element, and that element is isolated in*A*. - A space
*X*with the particular point topology, in particular the Sierpinski space, is scattered. This is an example of a scattered space that is not a T_{1}space. - The closure of a scattered set is not necessarily scattered. For example, in the Euclidean plane take a countably infinite discrete set
*A*in the unit disk, with the points getting denser and denser as one approaches the boundary. For example, take the union of the vertices of a series of n-gons centered at the origin, with radius getting closer and closer to 1. Then the closure of*A*will contain the whole circle of radius 1, which is dense-in-itself.

- In a topological space
*X*the closure of a dense-in-itself subset is a perfect set. So*X*is scattered if and only if it does not contain any nonempty perfect set. - Every subset of a scattered space is scattered. Being scattered is a hereditary property.
- Every scattered space
*X*is a T_{0}space. (*Proof:*Given two distinct points*x*,*y*in*X*, at least one of them, say*x*, will be isolated in . That means there is neighborhood of*x*in*X*that does not contain*y*.) - In a T
_{0}space the union of two scattered sets is scattered.^{ [3] }^{ [4] }Note that the T_{0}assumption is necessary here. For example, if with the indiscrete topology, and are both scattered, but their union, , is not scattered as it has no isolated point. - Every T
_{1}scattered space is totally disconnected.

- (
*Proof:*If*C*is a nonempty connected subset of*X*, it contains a point*x*isolated in*C*. So the singleton is both open in*C*(because*x*is isolated) and closed in*C*(because of the T_{1}property). Because*C*is connected, it must be equal to . This shows that every connected component of*X*has a single point.)

- Every second countable scattered space is countable.
^{ [5] } - Every topological space
*X*can be written in a unique way as the disjoint union of a perfect set and a scattered set.^{ [6] }^{ [7] } - Every second countable space
*X*can be written in a unique way as the disjoint union of a perfect set and a countable scattered open set.

- (
*Proof:*Use the perfect + scattered decomposition and the fact above about second countable scattered spaces, together with the fact that a subset of a second countable space is second countable.) - Furthermore, every closed subset of a second countable
*X*can be written uniquely as the disjoint union of a perfect subset of*X*and a countable scattered subset of*X*.^{ [8] }This holds in particular in any Polish space, which is the contents of the Cantor–Bendixson theorem.

- ↑ Steen & Seebach, p. 33
- ↑ Engelking, p. 59
- ↑ 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. - ↑ https://math.stackexchange.com/questions/3854864
- ↑ https://math.stackexchange.com/questions/376116
- ↑ Willard, problem 30E, p. 219
- ↑ https://math.stackexchange.com/questions/3856152
- ↑ https://math.stackexchange.com/questions/742025

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

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 subset of a topological space is called **nowhere dense** or **rare** if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered anywhere. For example, the integers are nowhere dense among the reals, whereas an open ball is not.

The **Baire category theorem** (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

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 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 fields of general topology and descriptive set theory, a **meagre set** is a set that, considered as a subset of a topological space, is in a precise sense small or negligible. A topological space T is called **meagre** if it is a meager subset of itself; otherwise, it is called **nonmeagre**.

In the mathematical field of topology, a **G _{δ} set** is a subset of a topological space that is a countable intersection of open sets. The notation originated in German with

In mathematics, a **Lindelöf space** is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of *compactness*, which requires the existence of a *finite* subcover.

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, 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 mathematics, **set-theoretic topology** is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC).

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 set and *p* ∈ *X*. The collection

In the mathematical field of topology, a **hyperconnected space** or **irreducible space** is a topological space *X* that cannot be written as the union of two proper closed sets. The name *irreducible space* is preferred in algebraic geometry.

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 *F*_{i} of closed subsets of there exists a pairwise disjoint family of open sets *U*_{i}, such that *F*_{i} ⊆ *U*_{i}. 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 demands that the above *U*_{i} are themselves a discrete family, which is stronger than pairwise disjoint.

In mathematics, particularly topology, a **G _{δ} space** is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A G

In topology and related areas of mathematics, a subset *A* of a topological space *X* is called **dense** if every point *x* in *X* either belongs to *A* or is a limit point of *A*; that is, the closure of *A* constitutes the whole set *X*. Informally, for every point in *X*, the point is either in *A* or 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.

- Engelking, Ryszard,
*General Topology*, Heldermann Verlag Berlin, 1989. ISBN 3-88538-006-4 - Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978].
*Counterexamples in Topology*(Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446. - Willard, Stephen (2004) [1970],
*General Topology*(Dover reprint of 1970 ed.), Addison-Wesley

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.