= Homotopy theory
Homotopy theory studies <topological spaces> and <continuous maps> through continuous deformation. A <homotopy> from $f:X\to Y$ to $g:X\to Y$ is a <continuous map> $H:X\times[0,1]\to Y$ with $H(x,0)=f(x)$ and $H(x,1)=g(x)$. Replacing a space by a <homotopy equivalent> one preserves its deformation-level information, including its <homotopy groups>.
For example, the punctured plane retracts onto its unit <circle> through
$$
H(x,t)=\left(1-t+\frac t{\|x\|}\right)x,\qquad x\in\mathbb R^2\setminus\{0\}.
$$
The coefficient stays positive, so the deformation never crosses the removed origin, and points already on the unit <circle> remain fixed. This is a <deformation retraction>.
The same viewpoint studies loops, higher-dimensional spheres and maps between spaces. It retains obstructions that survive continuous deformation while allowing distances, angles and the thickness of a region to change. A <homotopy equivalence> is weaker than a <homeomorphism>.
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