Subspace Topology

The subspace topology is the standard way to give a subset of a topological space its own topology using the structure of the surrounding, or ambient, space. Its open sets are exactly the intersections of the subset with ambient open sets. In topology, this construction allows subsets to be studied as spaces in their own right, while retaining a precise relationship with their surroundings. (pi.math.cornell.edu)

Definition and bases

Let $(X,\tau)$ be a topological space and $Y\subseteq X$. The subspace topology on $Y$ is

A member of $\tau_Y$ is called an open set in $Y$, or a relatively open subset of $Y$. Equipped with this topology, $Y$ is a subspace of $X$. (pi.math.cornell.edu)

The topology axioms follow from elementary set identities: intersecting $Y$ with the empty set and $X$ gives $\varnothing$ and $Y$; intersection with $Y$ commutes with arbitrary unions and finite intersections. If $\mathcal B$ is a basis for $\tau$, then

is a basis for $\tau_Y$. Empty members may be omitted. Thus an ambient basis suffices to describe every relatively open set. (pi.math.cornell.edu)

Openness depends on the space under consideration. For instance, in $Y=[0,1]\subseteq\mathbb R$,

is open in $Y$, although it is not open in the usual topology of the real line. The endpoint $0$ has relatively open neighborhoods that need not extend to negative numbers within $Y$. (math.toronto.edu)

Closed sets, closure, and neighborhoods

A subset $A\subseteq Y$ is a closed set in $Y$ precisely when

for some closed subset $F$ of $X$. Relative closedness therefore does not generally imply ambient closedness. However, if $Y$ is closed in $X$, every relatively closed subset of $Y$ is closed in $X$. The corresponding statement holds for open subsets when $Y$ is open in $X$. (pi.math.cornell.edu)

For $A\subseteq Y$, its closure satisfies

Consequently, closure retains precisely those ambient closure points that belong to the subspace. For example, the ambient closure of $(0,1)$ in $\mathbb R$ is $[0,1]$, whereas its closure in $Y=(0,1]$ is $(0,1]$. This example follows directly from the closure formula. (pi.math.cornell.edu)

Intersections $Y\cap N$, where $N$ ranges over ambient neighborhoods of $y\in Y$, form a neighborhood system in $Y$. It follows that a sequence of points of $Y$ converges to $y\in Y$ in the subspace exactly when it converges to $y$ in $X$. An ambient limit outside $Y$ is not a limit in the subspace. (math.mit.edu)

Continuity and the universal property

The inclusion $i:Y\to X$, defined by $i(y)=y$, is a continuous map, because

for every open $U\subseteq X$. The subspace topology is the coarsest topology on $Y$ making this inclusion continuous: any such topology must contain all these inverse images. (math.mit.edu)

Its universal property gives a stronger characterization. For any topological space $Z$ and function $f:Z\to Y$,

Here $i\circ f$ denotes function composition. In the reverse direction, each relatively open set has the form $Y\cap U$, and $f^{-1}(Y\cap U)=(i\circ f)^{-1}(U)$. Thus continuity into a subspace can be checked in its ambient space. (math.toronto.edu)

A topological embedding is a map that is a homeomorphism onto its image equipped with the subspace topology. Merely being continuous and injective is insufficient: the inverse on the image must also be continuous. This distinguishes a faithful realization of a space inside another from a map that preserves continuity only in one direction. (uni-math.gwdg.de)

Metric interpretation and examples

If $X$ is a metric space with metric $d$, restricting $d$ to $Y\times Y$ produces a metric whose topology is the subspace topology. Its open balls satisfy

The construction therefore agrees with the usual practice of measuring distances in a subset using the ambient metric. (math.hws.edu)

The integers form a discrete subspace of $\mathbb R$: an interval of radius less than $1/2$ around an integer intersects the integers only at that point. By contrast, the rational numbers are not discrete, since every interval around a rational contains other rationals. In

each nonzero point is isolated, but $0$ is not. This shows that a subspace can contain both isolated and nonisolated points. (math.toronto.edu)

In Euclidean space, a circle receives its usual topology by intersection with ambient open sets; small relatively open neighborhoods are arcs. “Subspace” here means a topological subset, not necessarily a linear subspace closed under vector operations. (math.toronto.edu)

Inherited properties and iterated subspaces

Every subspace of a Hausdorff space is Hausdorff: intersect disjoint ambient neighborhoods with the subspace. First and second countability are likewise inherited. Compactness is not inherited by arbitrary subsets, although every closed subspace of a compact space is compact. Connectedness is also not generally inherited; for example, the connected real line contains the disconnected subspace $\{0,1\}$. (math.toronto.edu)

The construction is transitive. If $A\subseteq Y\subseteq X$, the topology on $A$ induced through $Y$ equals the topology induced directly from $X$, since

Thus taking successive subspaces introduces no additional topological structure. (math.toronto.edu)