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Subspace topology

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).

Contents

1 Definition

2 Examples

3 Properties

4 Preservation of topological properties

5 See also

Definition

Given a topological space $(X,τ)$ and a subset $S\sube X$ , 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 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.

If $S$ is open, closed or dense in $(X,τ)$ we call $(S,τ S)$ an open subspace, closed subspace or dense subspace of $(X,τ)$ .

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.

More generally, suppose $i : S \to X$ 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.

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.

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 iff the composite map $i\circ f$ is continuous.

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 $S$ 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 $B_S = \{U\cap S : U \in B\}$ 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.

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.