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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, or the induced topology). Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
Euclid, detail from The School of Athens by Raphael. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...
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. IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...
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 topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. ...
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.
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
A natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...
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 or quotient 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. IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...
This property is characteristic in the sense that it can be used to define the subspace topology on Y. Image File history File links Subspace-01. ...
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 where a notion of distance between elements of the set 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 whenever a topological space has a certain topological property we have that all of its subspaces share the same property, then we say the topological property is hereditary. If only closed subspaces must 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 mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
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 subset of Euclidean space Rn is called compact if it is closed and bounded. ...
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 totally bounded space, or precompact space, is a space that can be covered by finitely many subsets of any fixed size. ...
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. ...
In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ...
In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ...
References - Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)
- Steen, Lynn A. and Seeback, J. Arthur Jr., Counterexamples in Topology, Holt, Rinehart and Winston (1970) ISBN 0030794854.
- Wilard, Stephen. General Topology, Dover Publications (2004) ISBN 0486434796
Counterexamples in Topology (1970) is a mathematics book by topologists Lynn A. Steen and J. Arthur Seebach, Jr. ...
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