Subspace topology - Citizendia

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, or the trace topology). Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

1 Definition
2 Examples
3 Properties
4 Preservation of topological properties
5 References
6 See also

Definition

Given a topological space $(X,τ)$ and a subset $S$ of $X$ , the subspace topology on $S$ is defined by

$\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace.$

That is, a subset of $S$ is open in the subspace topology if and only if it is the intersection of $S$ with an open set in $(X,τ)$ . ↔ 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 In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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.

If $S$ is open, closed or dense in $(X,τ)$ we call $(S,τ S)$ an open subspace, closed subspace or dense subspace of $(X,τ)$ , respectively. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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

Alternatively we can define the subspace topology for a subset $S$ of $X$ as the coarsest topology for which the inclusion map

$\iota: S \hookrightarrow X$

is continuous. 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 In Mathematics, if A is a Subset of B, then the inclusion map (also inclusion function, or canonical injection) is the In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

More generally, suppose $i$ 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. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group

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. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated "
The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 is not open in Q). In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions
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.

Properties

The subspace topology has the following characteristic property. Let $Y$ be a subspace of $X$ and let $i : Y \to X$ be the inclusion map. Then for any topological space $Z$ a map $f : Z\to Y$ is continuous if and only if the composite map $i\circ f$ is continuous. ↔

Characteristic property of the subspace topology

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 $f:X\to Y$ is continuous the restriction to $S$ is continuous.
If $f:X\to Y$ is continuous then $f:X\to f(X)$ 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 $S$ if and only if it is open in $X$ .
Suppose $S$ is a closed subspace of $X$ . Then a subspace of $S$ is closed in $S$ if and only if it is closed in $X$ .
If $B$ is a base for $X$ then $B_S = \{U\cap S : U \in B\}$ is a basis for $S$ . In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets
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 metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a metric or distance function is a function which defines a Distance between 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. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is If only closed subspaces must share the property we call it weakly hereditary.

every open and every closed subspace of a topologically complete space is topologically complete

every open subspace of a Baire space is a Baire space

every closed subspace of a compact space is compact. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Mathematics, a Baire space is a Topological space which intuitively speaking is very large and has "enough" points for certain limit processes

being a Hausdorff space is hereditary

precompactness is hereditary

being totally disconnected is hereditary

first countability and second countability are hereditary

References

Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)
Steen, Lynn A. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology and related branches of Mathematics, a totally bounded space is a space that can be covered by finitely many Subsets of any In Topology and related branches of Mathematics, a totally disconnected space is a Topological space which is maximally disconnected in the sense that In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability " and Seeback, J. Arthur Jr. , Counterexamples in Topology, Holt, Rinehart and Winston (1970) ISBN 0-03-079485-4. Counterexamples in Topology (1970 2nd ed 1978 is a book on Mathematics by topologists Lynn Steen and J
Wilard, Stephen. General Topology, Dover Publications (2004) ISBN 0-486-43479-6

Contents

Definition

Examples

Properties

Preservation of topological properties

References

See also