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Homotopy Equivalence

(R)
$f\simeq f$
(S)
$f\simeq g \implies g\simeq f$
$\begin{aligned} \htmlClass{proof-H}H & \htmlClass{proof-H}{\colon f\To g}\\ \htmlClass{proof-K}K & \htmlClass{proof-K}{\colon g\To f}\\ \htmlClass{proof-1t}{K(x,t)} & \htmlClass{proof-1t}{= H(x, 1-t)} \end{aligned}$
(T)
$f\simeq g\simeq h\implies f\simeq h$
$\begin{aligned} \htmlClass{proof-J}{J\colon f} & \htmlClass{proof-J}{\To h}\\ \htmlClass{proof-cases}{J(x,t)} &\htmlClass{proof-cases}{= \begin{cases} H(x, 2t) & 0\le t\le \frac12\\ K(x, 2t-1) & \frac12\le t\le 1 \end{cases}} \end{aligned}$

\f,\,\g,\,\h \colon S^1 \to \R^2 \qquad\qquad \htmlData{from-first=demo/HK}{\left.\begin{array}{c} H\colon f\To g\\ K\colon g\To h \end{array}\right\} \leadsto J\colon f\To h}