# 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 topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

## Definition

$\tau _{S}=\lbrace S\cap U\mid U\in \tau \rbrace .$ 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 $(X,\tau )$ . If $S$ is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of $(X,\tau )$ . Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

$\iota :S\hookrightarrow X$ is continuous.

## Examples

In the following, R represents the real numbers with their usual topology.

• The subspace topology of the natural numbers, as a subspace of R, is the discrete topology.
• The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 for example is not an open set in Q). If a and b are rational, then the intervals (a, b) and [a, b] are respectively open and closed, but if a and b are irrational, then the set of all x with a < x < b is both open and closed.
• The set [0,1] as a subspace of R is both open and closed, whereas as a subset of R it is only closed.
• As a subspace of R, [0, 1] ∪ [2, 3] is composed of two disjoint open subsets (which happen also to be closed), and is therefore a disconnected space.
• Let S = [0, 1) be a subspace of the real line R. Then [0, 1/2) 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

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 $X$ .

## Preservation of topological properties

If a topological space having some topological property implies its subspaces have that property, then we say the property is hereditary. If only closed subspaces must share the property we call it weakly hereditary.