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Topologist's sine curve

In the branch of mathematics known as topology, the topologist's sine curve is an example that has a number of 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 topologists 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.

References

Lynn Arthur Steen and J. Arthur Steenbach, Jr., Counterexamples in Topology [ISBN 048668735X]

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