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9 10 show that in a hausdorff space every set

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Unformatted text preview: a single point x (i.e., {x}) is a closed set. 10 Examples of topological spaces For any set X, there are two trivial topologies: Tc = {, X} and Td = P(X). Td is the topology in which each point (considered as a subset) is open (and hence, every subset is open). It is called the discrete topology. Tc is sometimes called the concrete topology. On R, there is the usual topology. We start with open intervals (a, b) = {x | a < x < b}. An open set is then any set which is a union of open intervals. 11 On Rn, there is the usual topology. One way to get this is to begin with the open balls with center a and radius r, where a Rn can be any point in Rn, and r is any positive real number: Bn(a, r) = {x Rn | |x - a| < r}. An open set is then any set which is a union of open balls. 12 Exercises: Examples of topological spaces 1. Check that each of the examples actually is a topological space. 2. For k < n, we can consider Rk to be a subset of Rn. Show that the inherited subspace topology is the same as the usua...
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