The discussion centers on the topology (ℝ,τ) defined by τ={∅}∪{u⊆ℝ:{-π,π}⊆u}. It concludes that the set [-4.4] is not open in this topology, as it does not satisfy the definition of an open set, which requires that for every point in the set, there exists a neighborhood entirely contained within the set. Conversely, the interval (-3,3) is closed in (ℝ,τ) because its complement, which includes points outside the interval, is open in the given topology.
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What does this mean: "{-π,π}⊆u"? What is "π"? Does this simply mean that U is in the topology if and only if any time x is in U, -x is also? In that case, these questions are close to trivial!blbl said:consider (ℝ,τ) , where τ={∅}∪{u⊆ℝ:{-π,π}⊆u}.
Is [-4.4] open in (ℝ,τ) ? Justify your answer
Is (-3,3) closed in (ℝ,τ) ? Justify your answer