Topologist's Sine Curve

We have all been familiar with the 'sine' graph in our 10th-grade math. Nice and simple to plot, with no complications. However, a spin-off version of this graph exists, laying the foundations for topological spaces. Let S = {(x, sin 1/x): 0<x<=1}, then, S' ={(x, sin 1/x): 0<x<=1}U {0×[−1,1]} has the capacity to oscillate infinitely in a finite domain! This occurs as 1/x becomes larger as x approaches zero.

This curve is clearly connected on the real line; however, it is not path-connected. That means if we want to go from a certain point to (0,0), we can surely walk in its direction on the curve, but we would still be a long way from the origin because of infinite oscillations near it. It tries to settle down at the origin, but keeps spiraling into existential jitters instead.

Even though sin(1/x) is connected, it is not locally connected, as we cannot find connected open sets on the y-axis (recall S is the union of S with {0 x [-1,1]}). Consider making an open ball around a point (0,y), where y is non-zero. We can see that the open ball contains infinitely many 'lines' of the graph; however, none of them are connected.

The topologist's curve is a specialized mathematical object demonstrating “what can go wrong” if only global connectedness is considered, without attention to local or path properties.

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