Any subset of a space becomes a space in its own right. Its open sets are the traces the surrounding open sets leave on it, and nothing else has to be invented.

Slice a loaf and look at the cut face. The pattern on the face is whatever the pattern inside the loaf leaves behind at that plane; nothing is drawn on afresh.

Let $A\subseteq X$. The subspace topology $\tau_A$ is the collection of the sets $U\cap A$ for $U$ open in $X$. This is what lets a property of spaces be asked of a subset at all: the words later chapters define — compact, connected, Hausdorff — are properties of a space, and a subset is given them by being made into one.

• A set open in the subspace need not be open in the whole space. In $[0,1]$ the set $[0,\tfrac12)$ is open, being the trace of $(-\tfrac12,\tfrac12)$.
• If $A$ is itself open in $X$, then a set open in $A$ is open in $X$ as well, since the intersection of two open sets is open.
• A subset of the real line takes the traces of the intervals, which is why every statement about intervals carries over to it unchanged.
• The closed sets of the subspace are the traces of the closed sets of the space, by the same argument with complements.

$A$ becomes a space in its own right, and its open sets are the traces $U\cap A$ that the open sets of $X$ leave on it. Nothing has to be invented: a subspace inherits its topology whole.

Check the three axioms. The empty set is the trace of the empty set and $A$ is the trace of the whole space, so the first holds. For the other two the work is done by two identities: intersecting with $A$ after taking a union is the same as taking the union of the intersections, and the same is true of a finite intersection. In both cases the set being traced is open in $X$ by the axioms there, so the trace is a member of $\tau_A$ by definition.

Both required members are traces of open sets, so the first axiom holds.

A union of traces is the trace of the union, and that union is open in $X$ with no restriction on size.

A finite intersection of traces is the trace of a finite intersection, which is open in $X$.

All three axioms hold, so the traces are a topology and $A$ is a space in its own right.

An open set of the line, traced on $[0,1]$.

Open in $[0,1]$, not open in the line: no interval around 0 stays inside it.

Two open sets meeting, which the finite rule allows.

A subset turned into a space, with no new choices to make.

Tracing the members of a basis gives a basis for the subspace, which is usually how a subspace is described.