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).

Definition

Given a topological space

(X, \tau)

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, \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. If

S

is open, closed or dense in

(X, \tau)

we call

(S, \tau_S)

an open subspace, closed subspace or dense subspace of

(X, \tau)

. Alternatively we can define the subspace topology for a subset

S

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.

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.

being a Hausdorff space is hereditary

precompactness is hereditary

being totally disconnected is hereditary