Consider the following topologies on R: • Ti = the trivial topology • T2 = the standard topology • T3 = the finite complement topology • TA = the lower limit topology • To = the discrete topology Determine, for each of these topologies, which of the others it contains.

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Q: Prove that ℝ with lower limit topology is Hausdorff

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A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

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Q: 1. Let T and S be topologies in X. Show that SnT is also a topology in X.

A: 1.

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A: Connected means it cannot be written as union of disjoint open sets

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A: see below