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Well order the real numbers, let [tex] {a_n}_{n \in S_{\Omega} } [/tex] be the the singleton sets of odd numbers in the well order (i.e. skip a number, grab a number, skip a number, grab a number). Since the real numbers are [tex] T_1 [/tex] then singleton sets are closed. [tex] \mathbb{R} [/tex] as a topological space is closed by definition. Let [tex] U := \mathbb{R} \bigcap_{n \in S_{\omega} } {a_n} [/tex] . Then U is closed and hence [tex] \mathbb{R} \backslash U [/tex] is open.
Is my logic correct here? It just seemed strange to me that this set would be open. Does anyone know what this set would look like? Thanks.
Is my logic correct here? It just seemed strange to me that this set would be open. Does anyone know what this set would look like? Thanks.