Why is the topologist's sine curve connected but not locally connected near the origin?
The topologist's sine curve is a continuous function that is defined by the following equation:
f(x) = sin(1/x), if x ≠ 0
f(0) = 0
This function is connected, meaning that it is possible to travel from any point on the curve to any other point on the curve without leaving the curve. However, the function is not locally connected near the origin. This means that there is no open neighborhood of the origin that is connected.
To see why this is the case, consider the following sequence of points:
x_n = 1/nπ, for n = 1, 2, 3, ...
As n approaches infinity, the points xn approach the origin. However, the values of f(xn) oscillate between -1 and 1. This means that there is no open neighborhood of the origin that contains only points that are connected to each other.
Related Questions
- What is the difference between connectedness and local connectedness?
- Why is the topologist's sine curve not locally connected near the origin?
- What other examples of connected but not locally connected spaces are there?
- What are some applications of connectedness and local connectedness in mathematics?
- How can we determine whether a given space is connected or locally connected?
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