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In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology). Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ...
History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...
Definition
Given a topological space (X,τ) and a subset , the subspace topology on S is defined by That is, a subset of S is open in the subspace topology iff it is the intersection of S with an open set in (X,τ). If S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of (X,τ). Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P...
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
If S is open, closed or dense in (X,τ) we call (S,τS) an open subspace, closed subspace or dense subspace of (X,τ). In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In topology and related areas of mathematics a subset A of a topological space X is called dense (in X) if the only closed subset of X containing A is X itself. ...
Alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map In mathematics, inclusion is a partial order on sets. ...
is continuous. In topology, a continuous function is generally defined as one for which preimages of open sets are open. ...
More generally, suppose is an injection from a set S to a topological space X. Then the subspace topology on S is defined as the coarsest topology for which i is continuous. The open sets in this topology are precisely the ones of the form i − 1(U) for U open in X. S is then homeomorphic to its image in X (also with the subspace topology) and i is called a topological embedding. Injection has multiple meanings: In mathematics, the term injection refers to an injective function. ...
This word should not be confused with homomorphism. ...
In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...
Examples - Given the real numbers with the usual topology the subspace topology of the natural numbers, as a subspace of the real numbers, is the discrete topology.
- The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 is not open in Q).
- Let S = [0,1) be a subspace the real line R. Then [0,½) is open in S but not in R. Likewise [½, 1) is closed in S but not in R. S is both open and closed as a subset of itself but not as a subset of R.
The text or formatting below is generated by a template which has been proposed for deletion. ...
Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...
In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
Properties The subspace topology has the following characteristic property. Let Y be a subspace of X and let be the inclusion map. Then for any topological space Z a map is continuous iff the composite map is continuous. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P...
This property is characteristic in the sense that it can be used to define the subspace topology on Y.
We list some further properties of the subspace topology. In the following let S be a subspace of X.
- If is continuous the restriction to S is continuous.
- If is continuous then is continuous.
- The closed sets in S are precisely the intersections of S with closed sets in X.
- If A is a subspace of S then A is also a subspace of X with the same topology. In other words the subspace topology that A inherits from S is the same as the one it inherits from X.
- Suppose S is an open subspace of X. Then a subspace of S is open in S iff it is open in X.
- Suppose S is a closed subspace of X. Then a subspace of S is closed in S iff it is closed in X.
- If B is a base for X then is a basis for S.
- The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.
In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...
In mathematics, a metric space is a set (or space) where a distance between points is defined. ...
In mathematics a metric or distance is a function which assigns a distance to elements of a set. ...
Preservation of topological properties If a topological space has a certain topological property and every subspace shares the same property we say the topological property is hereditary. If only closed subspaces share the property we call it weakly hereditary. In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ...
- being a Hausdorff space is hereditary
In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ...
In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
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