In topology, a **clopen set** (a portmanteau of **closed-open set**) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of *open* and *closed* are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open *and* closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!"^{ [1] } emphasizing that the meaning of "open"/"closed" for *doors* is unrelated to their meaning for *sets* (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "door spaces" their name.

In any topological space the empty set and the whole space are both clopen.^{ [2] }^{ [3] }

Now consider the space which consists of the union of the two open intervals and of The topology on is inherited as the subspace topology from the ordinary topology on the real line In the set is clopen, as is the set This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.

Now let be an infinite set under the discrete metric –that is, two points have distance 1 if they're not the same point, and 0 otherwise. Under the resulting metric space, any singleton set is open; hence any set, being the union of single points, is open. Since the complement of any set is therefore closed, all sets in the metric space are clopen.

As a less trivial example, consider the space of all rational numbers with their ordinary topology, and the set of all positive rational numbers whose square is bigger than 2. Using the fact that is not in one can show quite easily that is a clopen subset of ( is *not* a clopen subset of the real line ; it is neither open nor closed in )

- A topological space is connected if and only if the only clopen sets are the empty set and
- A set is clopen if and only if its boundary is empty.
^{ [4] } - Any clopen set is a union of (possibly infinitely many) connected components.
- If all connected components of are open (for instance, if has only finitely many components, or if is locally connected), then a set is clopen in if and only if it is a union of connected components.
- A topological space is discrete if and only if all of its subsets are clopen.
- Using the union and intersection as operations, the clopen subsets of a given topological space form a Boolean algebra.
*Every*Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.

- ↑ Munkres 2000, p. 91.
- ↑ Bartle, Robert G.; Sherbert, Donald R. (1992) [1982].
*Introduction to Real Analysis*(2nd ed.). John Wiley & Sons, Inc. p. 348. (regarding the real numbers and the empty set in R) - ↑ Hocking, John G.; Young, Gail S. (1961).
*Topology*. NY: Dover Publications, Inc. p. 56. (regarding topological spaces) - ↑ Mendelson, Bert (1990) [1975].
*Introduction to Topology*(Third ed.). Dover. p. 87. ISBN 0-486-66352-3.Let be a subset of a topological space. Prove that if and only if is open and closed.

(Given as Exercise 7)

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 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.

In mathematics, a **topological space** is, roughly speaking, a geometrical space in which *closeness* is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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, **open sets** are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.

In mathematics, the **closure** of a subset *S* of points in a topological space consists of all points in *S* together with all limit points of *S*. The closure of *S* may equivalently be defined as the union of *S* and its boundary, and also as the intersection of all closed sets containing *S*. Intuitively, the closure can be thought of as all the points that are either in *S* or "near" *S*. A point which is in the closure of *S* is a point of closure of *S*. The notion of closure is in many ways dual to the notion of interior.

In geometry, topology, and related branches of mathematics, a **closed set** is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

In mathematics, the **real line**, or **real number line** is the line whose points are the real numbers. That is, the real line is the set **R** of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

In mathematics, specifically in topology, the **interior** of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an **interior point** of S.

In mathematics, a **base** or **basis** for the topology τ of a topological space (*X*, τ) is a family *B* of open subsets of *X* such that every open set of the topology is equal to a union of some sub-family of *B*. For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.

In topology and mathematics in general, the **boundary** of a subset *S* of a topological space *X* is the set of points which can be approached both from *S* and from the outside of *S*. More precisely, it is the set of points in the closure of not belonging to the interior of An element of the boundary of is called a **boundary point** of The term **boundary operation** refers to finding or taking the boundary of a set. Notations used for boundary of a set include and Some authors use the term **frontier** instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, *Metric Spaces* by E. T. Copson uses the term boundary to refer to Hausdorff's **border**, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term **residue**, which is defined as the intersection of a set with the closure of the border of its complement.

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, a topological space is said to be a **Baire space**, if for any given countable collection of closed sets with empty interior in , their union also has empty interior in . Equivalently, a locally convex space which is not meagre in itself is called a Baire space. According to Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of a Baire space. Bourbaki coined the term "Baire space".

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 topology, a topological space is called **simply connected** if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.

In mathematics, a **field of sets** is a mathematical structure consisting of a pair consisting of a set and a family of subsets of called an **algebra over ** that contains the empty set as an element, and is closed under the operations of taking complements in finite unions, and finite intersections.

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 **cylinder set** is a set in the standard basis for the open sets of the product topology; they are also a generating family of the cylinder σ-algebra, which in the countable case is the product σ-algebra.

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.

- Munkres, James R. (2000).
*Topology*(Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. - Morris, Sidney A. "Topology Without Tears". Archived from the original on 19 April 2013.

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