Bases for Subspaces

Section Bases for Subspaces

Recall that a basis

B

for a topological space is a collection of sets that generate all of the open sets through unions. If we have a basis

B

for a topological space

(X, τ),

and if

A

is a subspace of

X,

we might ask if we can find a basis

B_{A}

from

B

in a natural way.

Let

(X, τ)

be a topological space with basis

B,

and let

A

be a subspace of

X .

There is a natural candidate to consider as a basis

B_{A}

for

A .

How do you think we should define the elements in

B_{A} ?

Recall that a set

B

is a basis for a topological space

X

🔗
For each $x \in X,$ there is a set in $B$ that contains $x .$
🔗
If $x \in X$ is an element of $B_{1} \cap B_{2}$ for some $B_{1}, B_{2} \in B,$ then there is a set $B_{3} \in B$ such that $x \in B_{3} \subseteq B_{1} \cap B_{2} .$

Show that your set from (a) is a basis for the induced topology on

A .

Topology: An Inquiry-Based Approach