This is question 7 on page 83 in Chapter 13 of James R. Munkers’ Topology 2nd Edition textbook.

Question

Consider the following topologies on \(\mathbb{R}\):

  • \(\mathcal{T}_1\) = the standard topology
  • \(\mathcal{T}_2\) = the topology of \(\mathbb{R}_K\)
  • \(\mathcal{T}_3\) = the finite complement topology
  • \(\mathcal{T}_4\) = the upper limit topology, having all sets (a, b] as basis
  • \(\mathcal{T}_5\) = the topology having all sets \((-\infty, a) = \{x : x \lt a\}\) as basis

Which of these topologies contains the others?

My Contains

Blank spaces mean I didn’t get to it.

  \(\mathcal{T}_1\) \(\mathcal{T}_2\) \(\mathcal{T}_3\) \(\mathcal{T}_4\) \(\mathcal{T}_5\)
\(\mathcal{T}_1\) \(\mathcal{T}_1 \subset \mathcal{T}_1\) \(\mathcal{T}_2\not\subset\mathcal{T}_1\) \(\mathcal{T}_3\not\subset\mathcal{T}_1\) \(\mathcal{T}_4\not\subset\mathcal{T}_1\) \(\mathcal{T}_5\not\subset\mathcal{T}_1\)
\(\mathcal{T}_2\) \(\mathcal{T}_1 \subset \mathcal{T}_2\) \(\mathcal{T}_2 \subset \mathcal{T}_2\)   \(\mathcal{T}_4\not\subset\mathcal{T}_2\)  
\(\mathcal{T}_3\) \(\mathcal{T}_1\not\subset\mathcal{T}_3\)   \(\mathcal{T}_3 \subset \mathcal{T}_3\)   \(\mathcal{T}_5\not\subset\mathcal{T}_3\)
\(\mathcal{T}_4\) \(\mathcal{T}_1 \subset \mathcal{T}_4\) \(\mathcal{T}_2\not\subset\mathcal{T}_4\)   \(\mathcal{T}_4 \subset \mathcal{T}_4\)  
\(\mathcal{T}_5\) \(\mathcal{T}_1 \subset \mathcal{T}_5\)   \(\mathcal{T}_3\not\subset\mathcal{T}_5\)   \(\mathcal{T}_5 \subset \mathcal{T}_5\)

Extras

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