A subset downstairs is open exactly when its full inverse image is open upstairs. The example is deliberately concrete; it is a test of the statement, not a substitute for it.
The mathematical object
A topology \(\tau\subseteq2^X\) specifies which subsets of \(X\) are open. Continuity is defined by inverse images, so it composes without requiring coordinates or distances.
I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.
One explicit computation
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
Why the identity matters
The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.
Where it can fail
Compact, connected, path-connected, and Hausdorff are independent properties in general spaces. Metric-space intuition supplies implications only with extra hypotheses.
The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.