Topologist's sine curve

In the branch of mathematics known as topology, the topologist's sine curve is a topological space with several interesting properties that make it an important textbook example.

It can be defined as the graph of the function sin(1/x) on the half-open interval (0, 1], together with the origin, under the topology induced from the Euclidean plane:

As x approaches zero from the right, the magnitude of the rate of change of 1/x increases. This is why the frequency of the sine wave increases as one moves to the left in the graph.

The topologist's sine curve T is connected but neither locally connected nor path connected. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a path.

The space T is the continuous image of a locally compact space (namely, let V be the space {−1} ∪ (0, 1], and use the map f from V to T defined by f(−1) = (0,0) and f(x) = (x, sin(1/x)) for x > 0), but T is not locally compact itself.

The topological dimension of T is 1.

...
Wikipedia