The **cocountable topology** or **countable complement topology** on any set *X* consists of the empty set and all cocountable subsets of *X*, that is all sets whose complement in *X* is countable. It follows that the only closed subsets are *X* and the countable subsets of *X*.

Every set *X* with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of *X*. It is also T_{1}, as all singletons are closed.

If *X* is an uncountable set, any two open sets intersect, hence the space is not Hausdorff. However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique. Since compact sets in *X* are finite subsets, all compact subsets are closed, another condition usually related to Hausdorff separation axiom.

The cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

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 **Borel set** is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

In topology and related branches of mathematics, a **normal space** is a topological space *X* that satisfies **Axiom T _{4}**: every two disjoint closed sets of

In topology and related branches of mathematics, a topological space is called **locally compact** if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

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 the mathematical field of topology, the **Alexandroff extension** is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named for the Russian mathematician Pavel Alexandroff. More precisely, let *X* be a topological space. Then the Alexandroff extension of *X* is a certain compact space *X** together with an open embedding *c* : *X* → *X** such that the complement of *X* in *X** consists of a single point, typically denoted ∞. The map *c* is a Hausdorff compactification if and only if *X* is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the **one-point compactification** or **Alexandroff compactification**. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space, a much larger class of spaces.

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 general topology, a branch of mathematics, a collection *A* of subsets of a set *X* is said to have the **finite intersection property** (FIP) if the intersection over any finite subcollection of *A* is non-empty. It has the **strong finite intersection property** (SFIP) if the intersection over any finite subcollection of *A* is infinite.

In topology, a topological space with the **trivial topology** is one where the only open sets are the empty set and the entire space. Such spaces are commonly called **indiscrete**, **anti-discrete**, or **codiscrete**. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.

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

In topology, a **second-countable space**, also called a **completely separable space**, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the **second axiom of countability**. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.

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

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, 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 mathematics, a topological space *X* is said to be **limit point compact** or **weakly countably compact** if every infinite subset of *X* has a limit point in *X*. 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 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 is constituting 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.

In general topology, a branch of mathematics, the **Appert topology**, named for Antoine Appert (1934), is a topology on the set *X* = **Z**^{+} = {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**.

- 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*(See example 20)*.

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