Although not noted in the literature, topology definitely enters electromagnetic theory.

This can clearly be seen in the experiment involving a Windhurst machine. Imagine a sprocket wheel (as from a bicycle) mounted on a wooden platform. An electric charge resides on one tooth of the wheel. If you measure on the counter outside this moving wheel, you detect magnetism. But if you measure on the wheel, you detect no magnetism, only electricity.

This difference inside and outside a reference frame is definitely topological -- alhthough, to note again, this is ignored in the literature.

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In topology, the square, the circle, the triangle, the many-sided polygon are all topologically equivalent. Why? Because all of them separate space into an "inside" and an "outide", which is what matters in topology. On the other hand, a "figure-eight" is topologically different, since it separates space into two distinct "insides" and one "outside'.

We daily encounter this sort of toplogy in electricity. Connect two "electric" wires to poles of a battery and to leads of a small light bulb, with a switch in between. When the switch is closed, electricity flows through wires, switch and bulb, to light up the bulb. But what shape does the circuit have to be? A circle or triangle or square or rectangle or any genus-zero plane figure will achieve the same result. But a figure-eight circuit could induce capacitance effects across the touching wires! (What was connectivity-along-the-line becomes Neighbor-connectivity.)

Physics students study two "Kirchoff circuitry laws", which Gustav Kirchoff (1824-87) said he derived by topological reasoning.