Topologist's sine curve
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In the branch of mathematics known as topology, the topologist's sine curve is an example that has several interesting properties.
It can be defined as a subset of the Euclidean plane as follows. Let S be the graph of the function sin(1/x) over the interval (0, 1]. Now let T be S union {(0,0)}. Give T the subset topology as a subset of the plane. T has the following properties:
- It is connected but not locally connected or path connected.
- It is not locally compact, but it is the continuous image of a locally compact space. (Namely, let V be the space {−1} union the interval (0, 1], and use the map f from V to T defined by f(−1) = (0, 0) and f(x) = (x, sin(1/x)).)
Two variations of the topologist's sine curve have other interesting properties.
The closed topologist's sine curve can be defined by taking the same set S defined above, and adding to it the set {(0, y) | y is in the interval [−1, 1] }. It is closed, but has similar properties to the topologist's sine curve -- it too is connected but not locally connected or path-connected.
The extended topologist's sine curve can be defined by taking the topologist's sine curve and adding to it the set {(x, 1) | x is in the interval [0, 1] }. It is arc connected but not locally connected.
Image of the curve
This is a crude plot of the Topologist's sine curve. There are two important notes about this plot.
1. As x approaches zero, 1/x approaches infinity at an increasing rate. This is why the frequency of the sine wave appears to increase on the left side of the graph.
2. As x increases the curve asymptotically approaches zero.
References
- Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).