Exercise 13.2

Question

Consider the nine topologies on the set $X = \left\{a,b,c\right\}$ indicated in Example~1 of \S 12.
Compare them; that is, for each pair of topologies, determine whether they are comparable, and if so, which is finer.

Dan Whitman · Accepted Answer

We label each of the topologies in Figure~12.1 with an ordered pair $(i,j)$ where $1 \leq i,j \leq 3$, $i$ is the row, $j$ is the column, and $(1,1)$ is the upper left corner.
The following matrix lists which of each pair is \emph{finer}, or ``Inc'' if they are incomparable.
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|}
& $(1,1)$ & $(1,2)$ & $(1,3)$ & $(2,1)$ & $(2,2)$ & $(2,3)$ & $(3,1)$ & $(3,2)$ & $(3,3)$ \
\hline
$(1,1)$ & $=$ & $(1,2)$ & $(1,3)$ & $(2,1)$ & $(2,2)$ & $(2,3)$ & $(3,1)$ & $(3,2)$ & $(3,3)$ \
\hline
$(1,2)$ &  & $=$ & Inc & Inc & Inc & Inc & $(1,2)$ & $(3,2)$ & $(3,3)$ \
\hline
$(1,3)$ &  &  & $=$ & $(1,3)$ & Inc & $(2,3)$ & $(1,3)$ & Inc & $(3,3)$ \
\hline
$(2,1)$ &  &  &  & $=$ & Inc & $(2,3)$ & Inc & $(3,2)$ & $(3,3)$ \
\hline
$(2,2)$ &  &  &  &  & $=$ & Inc & Inc & Inc & $(3,3)$ \
\hline
$(2,3)$ &  &  &  &  &  & $=$ & $(2,3)$ & Inc & $(3,3)$ \
\hline
$(3,1)$ &  &  &  &  &  &  & $=$ & $(3,2)$ & $(3,3)$ \
\hline
$(3,2)$ &  &  &  &  &  &  &  & $=$ & $(3,3)$ \
\hline
$(3,3)$ &  &  &  &  &  &  &  &  & $=$ \
\hline
\end{tabular}
\end{center}
We know that $\subsetneq$ forms a strict partial order on these topologies.
So we can also list all the maximal simply ordered subsets, each in order:
\begin{gather*}
(1,1) \subsetneq (2,2) \subsetneq (3,3) \
(1,1) \subsetneq (3,1) \subsetneq (1,2) \subsetneq (3,2) \subsetneq (3,3) \
(1,1) \subsetneq (3,1) \subsetneq (1,3) \subsetneq (2,3) \subsetneq (3,3) \
(1,1) \subsetneq (2,1) \subsetneq (1,3) \subsetneq (2,3) \subsetneq (3,3) \
(1,1) \subsetneq (2,1) \subsetneq (3,2) \subsetneq (3,3)
\end{gather*}

	$(1, 1)$	$(1, 2)$	$(1, 3)$	$(2, 1)$	$(2, 2)$	$(2, 3)$	$(3, 1)$	$(3, 2)$	$(3, 3)$

$(1, 1)$	$=$	$(1, 2)$	$(1, 3)$	$(2, 1)$	$(2, 2)$	$(2, 3)$	$(3, 1)$	$(3, 2)$	$(3, 3)$

$(1, 2)$		$=$	Inc	Inc	Inc	Inc	$(1, 2)$	$(3, 2)$	$(3, 3)$

$(1, 3)$			$=$	$(1, 3)$	Inc	$(2, 3)$	$(1, 3)$	Inc	$(3, 3)$

$(2, 1)$				$=$	Inc	$(2, 3)$	Inc	$(3, 2)$	$(3, 3)$

$(2, 2)$					$=$	Inc	Inc	Inc	$(3, 3)$

$(2, 3)$						$=$	$(2, 3)$	Inc	$(3, 3)$

$(3, 1)$							$=$	$(3, 2)$	$(3, 3)$

$(3, 2)$								$=$	$(3, 3)$

$(3, 3)$									$=$

Exercise 13.2

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Exercise 13.2

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