Subspaces and relative openness

Try

Inside the interval [0,2], the set [0,1) is open in the relative topology. It equals the intersection of [0,2] with the ambient open interval (−1,1). A point at the endpoint zero needs only a neighborhood within the space being studied, not extra points outside that space.

This is not a contradiction with [0,1) failing to be open in the whole real line. Openness always refers to a specified ambient space.

Predict

Predict whether a subset that is closed in a subspace must be closed in the original space. The whole subspace is always relatively closed, so the example Y=(0,1) inside R already answers no.

Predict what changes if Y itself is closed in X. Then a relatively closed subset of Y is also closed in X, because it is an intersection of two ambient closed sets. The extra hypothesis does real work.

Experiment

Intersect the chain topology on a,b,c with Y consisting of b,c.

Ambient open	Intersection with Y
Empty set	Empty set
Singleton a	Empty set
Pair a,b	Singleton b
Whole X	Whole Y

Different ambient opens can produce the same relative open. The inherited topology has three distinct members, not four.

Understand

For Y⊆X, the subspace topology consists of sets Y∩U with U open in X. These form a topology: unions and finite intersections distribute through intersection with Y, while the ambient empty and whole sets give the relative endpoints.

The inclusion map i:Y→X is continuous because i⁻¹(U)=Y∩U. More generally, h:Z→Y is continuous exactly when i∘h:Z→X is continuous. One direction follows by composition. For the converse, every relative open is Y∩U, and its inverse image under h equals the inverse image of U under the composite. This proves the criterion using all relative opens, without assuming that Y itself is ambient-open.

Subspace formation is transitive. If Z⊆Y⊆X, a twice-inherited open has form Z∩(Y∩U)=Z∩U. Thus inheriting through Y or directly from X gives the same topology on Z.

Relative closed sets have the parallel form Y∩F with F closed in X, obtained by taking complements within Y. If Y is open in X, every relatively open subset of Y is ambient-open. If Y is closed, every relatively closed subset is ambient-closed. Without those hypotheses, only the intersection representation is guaranteed.

Apply

Examine the rational numbers as a subspace of the real line. No singleton is open: every ordinary interval around a rational contains another rational. Yet every singleton is relatively closed, because it is the intersection of Q with an ambient closed singleton.

Repair “A subset of a discrete-looking point cloud is discrete.” The rational numbers may be drawn as separate plotted points, but their inherited neighborhoods contain infinitely many other rational points. A drawing with finitely many samples changes the object and cannot establish the subspace topology of Q.

Continue

Closure and boundary in a subspace will be computed next through neighborhood intersections. Optional depth: compare the relative topology on the integers with that on Q. A sufficiently short real interval isolates each integer, so the integers are discrete as a subspace, despite both sets being countable subsets of R.