Partitions of the reals such that closures of partition elements are saturated

Question

Suppose we have a partition $P$ of a set $S$ and a unary set operation $u:\mathscr{P}(S)\to\mathscr{P}(S)$ such that for each $A\in P$ the set $uA$ is saturated with respect to $P$ (a union of elements in $P$).

Can anyone suggest any references on this topic? So far the closest thing I have found is a 2013 paper by Christian Ronse entitled "Closures on partial partitions from closures on sets."

Here is the specific question I am curious about:

Conjecture. Let $X$ be a connected finite topological space. Let $u$ be the closure operation on $\mathbb{R}$ under the usual topology. Then there exists a partition $P$ of $\mathbb{R}$ satisfying the condition stated above, such that the quotient $\mathbb{R}/P$ is homeomorphic to $X$.

Even if this an intractable problem, it can be fun to search for counterexamples. Certain spaces are difficult to find a corresponding real partition (of the type above) for; some have had me thinking they were counterexamples for sure, then suddenly I find a partition that works.

Can anyone suggest any leads?

Added 22 July 2017

Taking into account the path (described below) that led me to the conjecture, instead of restricting to just one unary operation $u$ that is arbitrarily defined on $\mathscr{P}(S)\setminus P$, it might be better to allow any given collection of operations (of whatever arity), some or all of which may be required to satisfy various axioms (such as for example, closure axioms), with the partition condition applying to each.

But the meat of this post is supposed to be the conjecture, so if the above framework is too general to be of any use (it may be), then let's just scrap it.

I arrived at the conjecture as follows.

Back in grad school in the mid-1980s, the Kuratowski closure-complement-intersection problem (a good discussion can be found here) attracted my attention. I noticed that one could easily write a program to generate families of sets of reals under the usual topology (from a single seed set) by partitioning and working solely within the partition—provided it satisfies the above condition for the closure operation on $\mathbb{R}$ (the condition holds trivially for the complement operation).

In 2009 I posted a Javascript app here that lets users vary the seed set in a ten-point quotient of the reals to see what family it generates under closure and complement. About a year ago, I became interested in the actual space “under the hood” of this app.

A natural question arose: since connected seven-point spaces exist that contain Kuratowski 14-sets, might it be possible to replace my ten-point quotient with a seven-point one? (The answer is yes.) Further exploration led me to the conjecture, which I am currently about halfway finished verifying for all 94 connected five-spaces (it holds for all smaller ones). This is where I stop...there are too many connected six-spaces!

As one would guess, Cantor sets play a prominent role in many of the partitions (roughly half).

Added 5 August 2017

Professors who like to occasionally throw curveballs at students on Ph.D. quals might find a few in my answer below. While some partitions are trivial and most are easy to find, others are more challenging. For example, here is a moderately difficult one:

Problem. Give an example of a partition $\mathscr{P}=\{F_1,F_2,G_1,G_2,G_3\}$ of $\mathbb{R}$ under the usual topology such that $\varnothing\not\in\mathscr{P}$, each $F_i$ is closed, each $G_i$ is open, and $\overline{G_i}=G_i\cup F_1\cup F_2$ for $i=1,2,3.$

Solution. See space number 19 in the list of five-point spaces in my answer.

All of the evidence so far points to the truth of the conjecture. Short of finding a proof, it might be interesting to look for counterexamples in weakened spaces, for example in some other infinite connected space besides $\mathbb{R}.$ Or it might be interesting to replace $\mathbb{R}$ with finite spaces.

One more question. Many spaces in my answer seem to require the “middle thirds” structure of the Cantor construction in their associated real partitions. Many other spaces clearly do not. Assuming that some do in fact require it, what is it about them that makes this so?

Answer 1

Every nonempty connected space of five or fewer points satisfies the conjecture. $\def\R{\mathbb{R}}\def\hc{\hfill\cr}\def\sm{\setminus}\def\lw{\leftarrow}\def\ts{\textstyle}\def\sp#1#2#3#4{\matrix{#1\hc #2\hc #3\hc #4\hc}}\def\nq{\hfill\newline\quad}\def\nqq{\hfill\newline\quad\quad}$

We say a nonempty partition of a topological space is compatible with the topology (or just compatible) if the closure of each partition element is saturated. (Note that we get an equivalent definition using interior instead of closure.)

For each nonempty connected space $X$ such that $|X|\leq5$ we present a compatible partition $P$ of $\R$ such that $\R/P$ is homeomorphic to $X$ (partitions of finite intervals will always be defined so they extend to $\R$ by adjoining translations).

Spaces are ordered so it is clear that no two are homeomorphic. The list is thus self-contained, assuming one knows how many non-homeomorphic connected spaces exist of cardinalities one through five: 1, 2, 6, 21, 94 (sequence A001928). A list of all topological spaces of four or fewer points (up to homeomorphism) appears here and a list all five-point spaces appears here.

Per the usual Cantor set construction, for any non-degenerate closed interval $J,$ let $G_1(J),G_2(J),\ldots$ denote the sequence of (unions of) open middle thirds of $J.$ Thus, for each $n\geq1$ the set $G_n(J)$ is a disjoint union of $2^{n-1}$ finite open intervals. Moreover $G_m(J)\cap G_n(J)=\varnothing$ for $m\neq n.$

Let $M_0(J)$ denote the set of endpoints of $J$ and $M_n(J)$ the set of endpoints of intervals in $G_n(J).$ Let

$\sp{M(J)}{G(J)}{F(J)}{F'(J)}\sp{=}{=}{=}{=}\sp{\bigcup_{n=0}^\infty M_n(J)}{\bigcup_{n=1}^\infty G_n(J)}{J\sm G(J)}{F(J)\sm M(J)}\sp{\rm\ (endpoints),}{\rm\ (open\ middle\ thirds),}{{\rm\ (Cantor\ set),\ and}}{\rm\ (non\mbox{-}endpoints).}$

For any disjoint union $U$ of like intervals and partition $P$ of this same type of interval, the expression “$U\{P\}$” shall refer to the partition of $U$ that puts a congruent copy of $P$ in each interval of $U.$ For example, the notation $G_2([0,1])\{A\lw(0,1);B\lw[1,2)\}$ represents $$\ts A\supset({2\over18},{3\over18})\cup({14\over18},{15\over18})\quad{\rm and}\quad\ts B\supset[{3\over18},{4\over18})\cup[{15\over18},{16\over18}).$$ The arrow symbol $\leftarrow$ serves to remind that sets within braces are generally not members of the final partition.

Given a non-degenerate interval $I$ and discrete set $S=\{s_1,s_2,\ldots\}\subset I,$ let $h_S^+(I)$ denote an arbitrary discrete set $\{t_{i,j}\}\subset I\sm S$ such that for each $i=1,2,\ldots$ we have $t_{i,j}\nearrow s_i.$ Define $h_S^-(I)$ similarly with $t_{i,j}\searrow s_i.$ Let $H^+([a,b))$ denote the following “halving decomposition” of $[a,b)$: $$H^+([a,b))=\bigcup_{n=0}^\infty\big[b-{1\over2^n}(b-a),\,\,b-{1\over2^{n+1}}(b-a)\big).$$ Define $H^-((a,b])$ similarly.

Given a disjoint union $V$ of non-degenerate intervals, let $W_1(V),W_2(V),\ldots$ denote an arbitrary finite disjoint family of subsets of $V,$ each dense in $V,$ whose union is all of $V.$ For any given partition, the size of this family (for any given $V)$ will simply be the largest subscript that appears.

Finite topological spaces appear as ordered lists of closures of singletons. For example the list

ace bde c d

means “the topological space $\{a,b,c,d\}$ such that $\overline{\{a\}}=\{a,c,e\},$ $\overline{\{b\}}=\{b,d,e\},$ $\overline{\{c\}}=\{c\},$ and $\overline{\{d\}}=\{d\}.$” (Note that a space is $T_0$ iff this list contains no repetitions.) These lists appear in increasing order of (a) their total length, (b) for any given length in (a), their number of one-point elements, (c) for any given number in (b), their number of two-point elements, etc.

Partition elements are denoted $A,B,\ldots$ where the homeomorphism sends $a$ to $A,$ etc.

Lastly, note that since quotients preserve connectedness, the conjecture cannot be satisfied by any disconnected space.

One-point connected space

1. a

   $A=\R$

Two-point connected spaces $\def\sy#1{#1&=\ts}\def\sk#1{#1&\supset\ts}\def\sp#1#2{\begin{align}\sy{A}#1&\\\sy{B}#2&\\\end{align}}\def\ss#1#2{\begin{align}\sk{A}#1&\\\sk{B}#2&\\\end{align}}$

1. ab b

   $\sp{\R\sm B}{\{0\}}$

2. ab ab

   $\sp{W_1(\R)}{W_2(\R)}$

Three-point connected spaces $\def\sy#1#2{\lower#1pt\hbox{$#2$}&=\ts}\def\sk#1#2{\lower#1pt\hbox{$#2$}&\supset\ts}\def\sp#1#2#3#4#5#6{\begin{align}\sy{#4}{A}#1&\\\sy{#5}{B}#2&\\\sy{#6}{C}#3&\\\end{align}}\def\ss#1#2#3#4#5#6{\left\{\begin{align}\sk{#4}{A}#1&\\\sk{#5}{B}#2&\\\sk{#6}{C}#3&\\\end{align}\right.}$

1. ac bc c

   $\sp{(-\infty,0)}{(0,\infty)}{\{0\}}{0}{0}{0}$

2. abc b c

   $[0,2)\ss{(0,1)\cup(1,2)}{\{0\}}{\{1\}}{0}{0}{0}$

3. abc bc c

   $[0,1)\ss{(0,1)\sm B}{h_{\{0\}}^-([0,1))}{\{0\}}{.3}{.3}{.3}$

4. abc bc bc

   $[0,1]\ss{G([0,1])}{M([0,1])}{F'([0,1])}{0}{0}{0}$

5. abc abc c

   $\sp{W_1(\R\sm C)}{W_2(\R\sm C)}{\{0\}}{.2}{.2}{.2}$

6. abc abc abc

   $\sp{W_1(\R)}{W_2(\R)}{W_3(\R)}{0}{0}{0}$

Four-point connected spaces $\def\sy#1#2{\lower#1pt\hbox{$#2$}&=\ts}\def\sk#1#2{\lower#1pt\hbox{$#2$}&\supset\ts}\def\sp#1#2#3#4#5#6#7#8{\begin{align}\sy{#5}{A}#1&\\\sy{#6}{B}#2&\\\sy{#7}{C}#3&\\\sy{#8}{D}#4&\\\end{align}}\def\ss#1#2#3#4#5#6#7#8{\left\{\begin{align}\sk{#5}{A}#1&\\\sk{#6}{B}#2&\\\sk{#7}{C}#3&\\\sk{#8}{D}#4&\\\end{align}\right.}$

1. ad bd cd d

   $[0,1]\ss{\bigcup_{n=0}^\infty G_{3n+1}([0,1])}{\bigcup_{n=0}^\infty G_{3n+2}([0,1])}{\bigcup_{n=0}^\infty G_{3n+3}([0,1])}{F([0,1])}{.3}{.3}{.3}{.3}$

2. acd bd c d

   $[0,3)\ss{(0,1)\cup(1,2)}{(2,3)}{\{1\}}{\{0,2\}}{0}{0}{0}{0}$

3. abcd b c d

   $[0,3)\ss{(0,1)\cup(1,2)\cup(2,3)}{\{0\}}{\{1\}}{\{2\}}{0}{0}{0}{0}$

4. acd bd cd d

   $[0,2)\ss{(0,1)\sm C}{(1,2)}{h_{\{0\}}^-([0,1))\cup h_{\{1\}}^+((0,1])}{\{0,1\}}{.2}{.2}{.2}{.2}$

5. acd bcd c d

   $[0,2)\ss{(0,1)}{(1,2)}{\{0\}}{\{1\}}{0}{0}{0}{0}$

6. abcd bd c d

   $[0,2)\ss{(0,1)\cup(1,2)\sm B}{h_{\{1\}}^+((0,1])}{\{0\}}{\{1\}}{0}{0}{0}{0}$

7. acd bcd cd d

   $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq C\lw\{1\}\}\hc \lower.2pt\hbox{$D$}\supset F([0,1])\hc}\right.$

8. ad bcd bcd d

   $[0,2)\ss{(0,1)}{W_1((1,2))}{W_2((1,2))}{\{0,1\}}{0}{0}{0}{0}$

9. abcd bd cd d

   $[0,1)\ss{(0,1)\sm(B\cup C)}{h_{\{0\}}^-([0,1))\sm C}{h_{\{1\}}^+((0,1])}{\{0\}}{0}{0}{0}{0}$

10. abcd b cd cd

    $[0,2]\ss{G([0,2])\sm B}{\{1\}}{M([0,2])}{F'([0,2])}{0}{0}{0}{0}$

11. abcd bcd c d

    $[0,2)\ss{(0,1)\cup(1,2)\sm B}{h_{\{0\}}^-([0,1))\cup h_{\{1\}}^-([1,2))}{\{0\}}{\{1\}}{.3}{.3}{.2}{.2}$

12. acd bcd cd cd

    $[0,1]\ss{\bigcup_{n=0}^\infty G_{2n+1}([0,1])}{\bigcup_{n=0}^\infty G_{2n+2}([0,1])}{M([0,1])}{F'([0,1])}{.3}{.3}{.3}{.3}$

13. abcd bcd cd d

    $[0,1)\ss{(0,1)\sm(B\cup C)}{h_C^-((0,1))}{h_{\{0\}}^-([0,1))}{\{0\}}{.3}{.3}{.3}{.2}$

14. abcd abcd c d

    $[0,2)\ss{W_1((0,1)\cup(1,2))}{W_2((0,1)\cup(1,2))}{\{0\}}{\{1\}}{0}{0}{0}{0}$

15. abcd bcd cd cd

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq B\lw\{1\}\}\hc \lower.2pt\hbox{$C$}\supset M([0,1])\hc \lower.2pt\hbox{$D$}\supset F'([0,1])\hc}\right.$

16. abcd bcd bcd d

    $[0,1]\ss{G([0,1])}{\bigcup_{n=1}^\infty M_n([0,1])}{F'([0,1])}{M_0([0,1])}{0}{0}{0}{0}$

17. abcd abcd cd d

    $[0,1)\ss{W_1((0,1))\sm C}{W_2((0,1))\sm C}{h_{\{0\}}^-([0,1))}{\{0\}}{0}{0}{0}{0}$

18. abcd abcd cd cd

    $[0,1]\ss{W_1(G([0,1]))}{W_2(G([0,1]))}{M([0,1])}{F'([0,1])}{0}{0}{0}{0}$

19. abcd bcd bcd bcd

    $[0,1]\ss{G([0,1])}{\bigcup_{n=0}^\infty M_{2n}([0,1])}{\bigcup_{n=0}^\infty M_{2n+1}([0,1])}{F'([0,1])}{.3}{.3}{.3}{.3}$

20. abcd abcd abcd d

    $\sp{W_1(\R\sm D)}{W_2(\R\sm D)}{W_3(\R\sm D)}{\{0\}}{.2}{.2}{.2}{.2}$

21. abcd abcd abcd abcd

    $\sp{W_1(\R)}{W_2(\R)}{W_3(\R)}{W_4(\R)}{0}{0}{0}{0}$

Five-point connected spaces $\def\sy#1{#1&=\ts}$ $\def\sk#1{#1&\supset\ts}$ $\def\sp#1#2#3#4#5{\begin{align}\sy{A}#1&\\\sy{B}#2&\\\sy{C}#3&\\\sy{D}#4&\\\sy{E}#5&\\\end{align}}$ $\def\ss#1#2#3#4#5{\left\{\begin{align}\sk{A}#1&\\\sk{B}#2&\\\sk{C}#3&\\\sk{D}#4&\\\sk{E}#5&\\\end{align}\right.}$

1. ae be ce de e

   $[0,1]\ss{\bigcup_{n=0}^\infty G_{4n+1}([0,1])}{\bigcup_{n=0}^\infty G_{4n+2}([0,1])}{\bigcup_{n=0}^\infty G_{4n+3}([0,1])}{\bigcup_{n=0}^\infty G_{4n+4}([0,1])}{F([0,1])}$

2. ade bd ce d e

   $[0,4)\ss{(0,1)\cup(2,3)}{(1,2)}{(3,4)}{\{1,2\}}{\{0,3\}}$

3. ade be ce d e

   $[0,2]\ss{\bigcup_{n=0}^\infty G_{3n+1}([0,2])\sm D}{\bigcup_{n=0}^\infty G_{3n+2}([0,2])}{\bigcup_{n=0}^\infty G_{3n+3}([0,2])}{\{1\}}{F([0,2])}$

4. ace bde c d e

   $[0,4)\ss{(0,1)\cup(1,2)}{(2,3)\cup(3,4)}{\{1\}}{\{3\}}{\{0,2\}}$

5. acde be c d e

   $[0,4)\left\{\matrix{H^-((0,1])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq E\lw\{1,2\}\}\hc\hskip-2pt\begin{align}A&\supset(1,2)\cup(2,3)\cup(3,4)&\\ C&\supset\{2\}&\\ D&\supset\{3\}&\\ E&\supset\{0\}&\end{align}\hc}\right.$

6. abcde b c d e

   $[0,4)\ss{(0,1)\cup(1,2)\cup(2,3)\cup(3,4)}{\{0\}}{\{1\}}{\{2\}}{\{3\}}$

7. ade be ce de e

   $[0,1]\left\{\matrix{G_{3n+1}([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq D\lw\{1\}\}{\rm\ for\ }n=0,1,2,\ldots\hc\hskip-1pt\begin{align}\sk{B}\bigcup_{n=0}^\infty G_{3n+2}([0,1])&\\\sk{C}\bigcup_{n=0}^\infty G_{3n+3}([0,1])&\\\sk{E} F([0,1])&\end{align}\hc}\right.$

8. ade bde ce d e

   $[0,1]\left\{\matrix{G_1([0,1])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq D\lw\{1\}\}\hc\hskip-1pt\begin{align}\sk{A}\bigcup_{n=0}^\infty G_{3n+2}([0,1])&\\\sk{B}\bigcup_{n=0}^\infty G_{3n+3}([0,1])&\\\sk{C}\bigcup_{n=1}^\infty G_{3n+1}([0,1])&\\\sk{E} F([0,1])&\end{align}\hc}\right.$

9. ace bde ce d e

   $[0,1]\left\{\matrix{G_{2n+1}([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq C\lw\{1\}\}{\rm\ for\ }n=0,1,2,\ldots\hc G_2([0,1])\{\nq B\lw(0,1)\cup(1,2);\nq D\lw\{1\}\}\hc\hskip-1pt\begin{align}\sk{B}\bigcup_{n=1}^\infty G_{2n+2}([0,1])&\\\sk{E} F([0,1])&\end{align}\hc}\right.$

10. acde be ce d e

    $[0,1]\left\{\matrix{G_{2n+1}([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq C\lw\{1\}\}{\rm\ for\ }n=1,2,\ldots\hc G_1([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq D\lw\{1\}\}\hc\hskip-1pt\begin{align}\sk{B}\bigcup_{n=0}^\infty G_{2n+2}([0,1])&\\\sk{E} F([0,1])&\end{align}\hc}\right.$

11. acde bd ce d e

    $[0,3)\ss{(0,1)\cup(1,2)\sm C}{(2,3)}{h^-_{\{1\}}([1,2))}{\{0,2\}}{\{1\}}$

12. acde bde c d e

    $[0,3)\ss{(0,1)\cup(1,2)}{(2,3)}{\{1\}}{\{0\}}{\{2\}}$

13. abcde be c d e

    $[0,3)\ss{(0,1)\cup(1,2)\cup(2,3)\sm B}{h^-_{\{2\}}([2,3))}{\{0\}}{\{1\}}{\{2\}}$

14. ade bde ce de e

    $[0,1]\left\{\matrix{G_{2n+1}([0,1])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq D\lw\{1\}\}{\rm\ for\ }n=0,1,2,\ldots\hc\hskip-2pt\begin{align}\sk{C}\bigcup_{n=0}^\infty G_{2n+2}([0,1])&\\\sk{E}F([0,1])&\end{align}\hc}\right.$

15. ace bde ce de e

    $[0,1]\left\{\matrix{G_{2n+1}([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq C\lw\{1\}\}{\rm\ for\ }n=0,1,2,\ldots\hc G_{2n+2}([0,1])\{\nq B\lw(0,1)\cup(1,2);\nq D\lw\{1\}\}{\rm\ for\ }n=0,1,2,\ldots\hc\hskip-2pt\begin{align}\sk{E}F([0,1])&\end{align}\hc}\right.$

16. ae be cde cde e

    $[0,1]\left\{\matrix{\lower.2pt\hbox{$A$}\supset\bigcup_{n=0}^\infty G_{3n+1}([0,1])\hc\lower.2pt\hbox{$B$}\supset\bigcup_{n=0}^\infty G_{3n+2}([0,1])\hc G_{3n+3}([0,1])\{\nq C\lw W_1((0,1));\nq D\lw W_2((0,1))\}{\rm\ for\ }n=0,1,2,\ldots\hc \lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

17. acde be ce de e

    $[0,1]\left\{\matrix{G_{2n+1}([0,1])\{\nq A\lw(0,1)\cup(1,2)\cup(2,3);\nq C\lw\{1\};\nq D\lw\{2\}\}{\rm\ for\ }n=0,1,2,\ldots\hc\hskip-2pt\begin{align}\sk{B}\bigcup_{n=0}^\infty G_{2n+2}([0,1])&\\\sk{E}F([0,1])&\end{align}\hc}\right.$

18. acde bc c de de

    $[0,4)\ss{(0,1)\cup G([1,2])\cup(2,3)}{(3,4)}{\{0,3\}}{M([1,2])}{F'([1,2])}$

19. ade bde cde d e

    $[0,4)\ss{\bigcup_{n=0}^\infty \big(G_{3n+1}([0,1])\cup G_{3n+1}([2,3])\big)\cup(1,2)\cup(3,4)}{\bigcup_{n=0}^\infty \big(G_{3n+2}([0,1])\cup G_{3n+2}([2,3])\big)}{\bigcup_{n=0}^\infty \big(G_{3n+3}([0,1])\cup G_{3n+3}([2,3])\big)}{F([0,1])}{F([2,3])}$

20. abe b cde cde e

    $[0,2]\left\{\matrix{\lower.2pt\hbox{$A$}\supset\bigcup_{n=0}^\infty G_{2n+1}([0,2])\sm B\hc\lower.2pt\hbox{$B$}\supset\{1\}\hc G_{2n+2}([0,2])\{\nq C\lw W_1((0,1));\nq D\lw W_2((0,1))\}{\rm\ for\ }n=0,1,2,\ldots\hc \lower.2pt\hbox{$E$}\supset F([0,2])\hc}\right.$

21. acde bde ce d e

    $[0,2)\ss{(0,1)\sm C}{(1,2)}{h_{\{0\}}^-([0,1))}{\{1\}}{\{0\}}$

22. acde be cde d e

    $[0,3)\ss{(0,1)\cup(1,2)\sm C}{(2,3)}{h^-_{\{0\}}([0,1))\cup h^+_{\{1\}}((0,1])\cup h^+_{\{2\}}((1,2])}{\{1\}}{\{0,2\}}$

23. acde bde c de e

    $[0,4)\left\{\matrix{G([0,1])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq D\lw\{1\}\}\hc\hskip-2pt\begin{align}\sk{A}(1,2)\cup G([2,3])\cup(3,4)&\\\sk{C}F([2,3])\\\sk{E}F([0,1])&\end{align}\hc}\right.$

24. abcde be ce d e

    $[0,2)\ss{(0,1)\cup(1,2)\sm(B\cup C)}{h^+_{\{1\}}((0,1])}{h^-_{\{1\}}([1,2))}{\{0\}}{\{1\}}$

25. abcde bd ce d e

    $[0,2)\ss{(0,1)\cup(1,2)\sm(B\cup C)}{h^-_{\{0\}}([0,1))}{h^-_{\{1\}}([1,2))}{\{0\}}{\{1\}}$

26. abcde b c de de

    $[0,4)\ss{(0,1)\cup G([1,2])\cup(2,3)\cup(3,4)}{\{0\}}{\{3\}}{M([1,2])}{F'([1,2])}$

27. acde bcde c d e

    $[0,4)\ss{(0,1)\cup(2,3)}{(1,2)\cup(3,4)}{\{0\}}{\{1,3\}}{\{2\}}$

28. abcde bde c d e

    $[0,3)\ss{(0,1)\cup(1,2)\cup(2,3)\sm B}{h^-_{\{1\}}[1,2)\cup h^+_{\{2\}}(1,2]}{\{0\}}{\{1\}}{\{2\}}$

29. ade bde cde de e

    $[0,2)\left\{\matrix{H^-((0,1])\{\nq A\lw\bigcup_{n=0}^\infty G_{3n+1}([0,1]);\nq B\lw\bigcup_{n=0}^\infty G_{3n+2}([0,1]);\nq C\lw\bigcup_{n=0}^\infty G_{3n+3}([0,1]);\nq D\lw F([0,1])\sm\{0\}\}\hc\hskip-2pt\begin{align}\sk{A}(1,2)&\\\sk{E}\{0\}&\end{align}\hc}\right.$

30. abe be cde cde e

    $[0,2]\left\{\matrix{G_{2n+1}([0,2])\{\nq A\lw(0,1)\cup(1,2);\nq B\lw\{1\}\}{\rm\ for\ }n=0,1,2,\ldots\hc G_{2n+2}([0,2])\{\nq C\lw W_1((0,1));\nq D\lw W_2((0,1))\}{\rm\ for\ }n=0,1,2,\ldots\hc \lower.2pt\hbox{$E$}\supset F([0,2])\hc}\right.$

31. acde bde ce de e

    $[0,3]\left\{\matrix{G([0,3])\{\nq A\lw(0,1)\cup(1,2);\nq B\lw(2,3);\nq C\lw\{1\};\nq D\lw\{2\}\}\hc \lower.1pt\hbox{$E$}\supset F([0,3])\hc}\right.$

32. acde be cde de e

    $[0,1]\left\{\matrix{G_{2n+1}([0,1])\{\nq A\lw(0,1)\cup(1,2)\sm C;\nq C\lw h^-_{\{1\}}([1,2));\nq D\lw\{1\}\}{\rm\ for\ }n=0,1,2,\ldots\hc \lower.1pt\hbox{$B$}\supset\bigcup_{n=0}^\infty G_{2n+2}([0,1])\hc \lower.1pt\hbox{$E$}\supset F([0,1])\hc}\right.$

33. acde bde c de de

    $[0,2]\ss{\bigcup_{n=0}^\infty G_{2n+1}([0,2])\sm C}{\bigcup_{n=0}^\infty G_{2n+2}([0,2])}{\{1\}}{M([0,2])}{F'([0,2])}$

34. abcde be ce de e

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(1,2)\cup(2,3)\cup(3,4);\nq B\lw\{1\};\nq C\lw\{2\};\nq D\lw\{3\}\}\hc \lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

35. abcde bc c de de

    $[0,3)\ss{G([0,1])\cup(1,2)\cup(2,3)\sm B}{h^-_{\{2\}}([2,3))}{\{2\}}{M([0,1])}{F'([0,1])}$

36. acde bde cde d e

    $[0,4)\left\{\matrix{\big(G_{2n+1}([0,1])\cup G_{2n+1}([2,3])\big)\{\nq A\lw(0,1)\cup(1,2);\nq C\lw\{1\}\}{\rm\ for\ }n=0,1,2,\ldots\hc \begin{align}\sk{A}(1,2)\cup(3,4)&\\\sk{B}\bigcup_{n=0}^\infty\big(G_{2n+2}([0,1])\cup G_{2n+2}([2,3])\big)&\\\sk{D}F([0,1])&\\\sk{E}F([2,3])\end{align}\hc}\right.$

37. acde bcde ce d e

    $[0,1]\left\{\matrix{G_1([0,1])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq D\lw\{1\}\}\hc G_n([0,1])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq C\lw\{1\}\}{\rm\ for\ }n=2,3,\ldots\hc\lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

38. ae bcde bcde d e

    $[0,3)\ss{(1,2)}{W_1((0,1)\cup(2,3))}{W_2((0,1)\cup(2,3))}{\{0\}}{\{1,2\}}$

39. abcde bde ce d e

    $[0,2)\ss{(0,1)\cup(1,2)\sm(B\cup C)}{h^-_{\{0\}}([0,1))\cup h^+_{\{1\}}((0,1])}{h^-_{\{1\}}([1,2))}{\{0\}}{\{1\}}$

40. abcde bde c de e

    $[0,2)\ss{(0,1)\cup(1,2)\sm(B\cup D)}{h^-_D((1,2))}{\{0\}}{h^-_{\{1\}}([1,2))}{\{1\}}$

41. abcde bcde c d e

    $[0,3)\ss{(0,1)\cup(1,2)\cup(2,3)\sm B}{h^-_{\{0\}}([0,1))\cup h^-_{\{1\}}([1,2))\cup h^-_{\{2\}}([2,3))}{\{0\}}{\{1\}}{\{2\}}$

42. ade bde cde de de

    $[0,1]\ss{\bigcup_{n=0}^\infty G_{3n+1}([0,1])}{\bigcup_{n=0}^\infty G_{3n+2}([0,1])}{\bigcup_{n=0}^\infty G_{3n+3}([0,1])}{M([0,1])}{F'([0,1])}$

43. abcde bc bc de de

    $[0,4)\ss{G([0,1])\cup(1,2)\cup G([2,3])\cup(3,4)}{M([0,1])}{F'([0,1])}{M([2,3])}{F'([2,3])}$

44. abe abe cde cde e

    $[0,1]\left\{\matrix{G_{2n+1}([0,1])\{\nq A\lw W_1((0,1));\nq B\lw W_2((0,1))\}{\rm\ for\ }n=0,1,2,\ldots\hc G_{2n+2}([0,1])\{\nq C\lw W_1((0,1));\nq D\lw W_2((0,1))\}{\rm\ for\ }n=0,1,2,\ldots\hc\lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

45. acde bde cde de e

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\sm C;\nq B\lw(1,2);\nq C\lw h^+_{\{1\}}((0,1]);\nq D\lw\{1\}\}\hc\lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

46. acde be cde cde e

    $[0,1]\left\{\matrix{G_{2n+1}([0,1])\{\nq A\lw(0,1)\cup G([1,2])\cup(2,3);\nq C\lw M([1,2]);\nq D\lw F'([1,2])\}{\rm\ for\ }n=0,1,2,\ldots\hc\begin{align}\sk{B}\bigcup_{n=0}^\infty G_{2n+2}([0,1])&\\\sk{E}F([0,1])\end{align}\hc}\right.$

47. acde bcde ce de e

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(2,3);\nq B\lw(1,2);\nq C\lw\{1\};\nq D\lw\{2\}\}{\rm\ for\ }n=0,1,2,\ldots\hc\lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

48. acde bcde c de de

    $[0,4)\ss{\bigcup_{n=0}^\infty\big(G_{2n+1}([0,1])\cup G_{2n+1}([2,3])\big)\cup(1,2)\cup(3,4)}{\bigcup_{n=0}^\infty\big(G_{2n+2}([0,1])\cup G_{2n+2}([2,3])\big)}{F([0,1])}{M([2,3])}{F'([2,3])}$

49. ae bcde bcde de e

    $[0,2)\ss{(0,1)}{W_1((1,2))\sm D}{W_2((1,2))\sm D}{h^-_{\{1\}}([1,2))\cup h^+_{\{2\}}((1,2])}{\{0,1\}}$

50. abcde bde ce de e

    $[0,1)\ss{(0,1)\sm(B\cup C\cup D)}{h^+_D((0,1))}{h^-_{\{0\}}([0,1))\sm(B\cup D)}{h^+_{\{1\}}((0,1])}{\{0\}}$

51. abcde bde c de de

    $[0,3)\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq B\lw\{1\}\}\hc\hskip-1pt\begin{align}\sk{A}(1,2)\cup(2,3)&\\\sk{C}\{2\}&\\\sk{D}M([0,1])&\\\sk{E}F'([0,1])\end{align}\hc}\right.$

52. acde bcde cde d e

    $[0,4)\left\{\matrix{G([0,1])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq C\lw\{1\}\}\hc G([2,3])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq C\lw\{1\}\}\hc\hskip-1pt\begin{align}\sk{A}(1,2)\cup(3,4)&\\\sk{D}F([0,1])&\\\sk{E}F([2,3])\end{align}\hc}\right.$

53. ade bcde bcde d e

    $[0,2)\ss{(0,1)}{W_1((1,2))}{W_2((1,2))}{\{0\}}{\{1\}}$

54. abcde bde cde d e

    $[0,2)\ss{(0,1)\cup(1,2)\sm(B\cup C)}{h^-_{\{0\}}([0,1))\cup h^-_{\{1\}}([1,2))\sm C}{h^+_{\{1\}}((0,1])\cup h^+_{\{2\}}((1,2])}{\{0\}}{\{1\}}$

55. abcde b cde cde e

    $[0,2]\left\{\matrix{\lower.2pt\hbox{$A$}\supset\bigcup_{n=0}^\infty G_{2n+1}([0,2])\sm B\hc\lower.2pt\hbox{$B$}\supset\{1\}\hc G_{2n+2}([0,2])\{\nq A\lw(0,1)\cup G([1,2])\cup(2,3);\nq C\lw M([1,2]);\nq D\lw F'([1,2])\}{\rm\ for\ }n=0,1,2,\ldots\hc\lower.2pt\hbox{$E$}\supset F([0,2])\hc}\right.$

56. abcde bcde ce d e

    $[0,2)\ss{(0,1)\cup(1,2)\sm(B\cup C)}{h^-_{\{0\}}([0,1))\cup h^-_{C}((1,2))}{h^-_{\{1\}}([1,2))}{\{0\}}{\{1\}}$

57. abcde abcde c d e

    $[0,3)\ss{W_1((0,1)\cup(1,2)\cup(2,3))}{W_2((0,1)\cup(1,2)\cup(2,3))}{\{0\}}{\{1\}}{\{2\}}$

58. acde bde cde de de

    $[0,1]\left\{\matrix{G_{2n+1}([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq C\lw\{1\}\}{\rm\ for\ }n=0,1,2,\ldots\hc\begin{align}\sk{B}\bigcup_{n=0}^\infty G_{2n+2}([0,1])&\\\sk{D}M([0,1])&\\\sk{E}F'([0,1])\end{align}\hc}\right.$

59. acde bcde cde de e

    $[0,2)\left\{\matrix{H^-((0,1])\{\nq H^-((1,2])\{\nqq A\lw(0,1);\nqq B\lw(1,2);\nqq C\lw\{1,2\}\};\nq A\lw(0,1);\nq D\lw\{1\}\}\hc\lower.2pt\hbox{$A$}\supset(1,2)\hc\lower.2pt\hbox{$E$}\supset\{0\}\hc}\right.$

60. ade bcde bcde de e

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1);\nq B\lw W_1((1,2));\nq C\lw W_2((1,2));\nq D\lw\{1\}\}\hc\lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

61. abcde bde cde de e

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(1,2)\sm(B\cup C);\nq B\lw h^+_{\{1\}}((0,1]);\nq C\lw h^-_{\{1\}}([1,2));\nq D\lw\{1\}\}\hc\lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

62. abcde be cde cde e

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(1,2)\cup G([2,3])\cup(3,4);\nq B\lw\{1\};\nq C\lw M([2,3]);\nq D\lw F'([2,3])\}\hc\lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

63. abcde bcde ce de e

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(1,2)\cup(2,3)\sm B;\nq B\lw h^-_{\{1\}}([1,2))\cup h^-_{\{2\}}([2,3));\nq C\lw\{1\};\nq D\lw\{2\}\}\hc\lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

64. abcde bcde c de de

    $[0,1]\left\{\matrix{G_1([0,1])\{\nq A\lw(0,1)\cup(1,2)\sm B;\nq B\lw h^-_{\{1\}}([1,2));\nq C\lw\{1\}\}\hc G_n([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq B\lw\{1\}\}{\rm\ for\ }n=2,3,\ldots\hc\lower.2pt\hbox{$D$}\supset M([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

65. abcde bcde cde d e

    $[0,2)\ss{(0,1)\cup(1,2)\sm(B\cup C)}{h^-_{C\,\cap\,(0,1)}((0,1))\cup h^-_{C\,\cap\,(1,2)}((1,2))}{h^-_{\{0\}}([0,1))\cup h^-_{\{1\}}([1,2))}{\{0\}}{\{1\}}$

66. abcde abcde ce d e

    $[0,2)\ss{W_1((0,1)\cup(1,2))\sm C}{W_2((0,1)\cup(1,2))\sm C}{h^-_{\{1\}}([1,2))}{\{0\}}{\{1\}}$

67. acde bcde cde de de

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1);\nq B\lw(1,2);\nq C\lw\{1\}\}\hc\lower.2pt\hbox{$D$}\supset M([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

68. ade bcde bcde de de

    $[0,1]\left\{\matrix{G_{2n+2}([0,1])\{\nq B\lw W_1((0,1));\nq C\lw W_2((0,1))\}{\rm\ for\ }n=0,1,2,\ldots\hc\lower.2pt\hbox{$A$}\supset\bigcup_{n=0}^\infty G_{2n+1}([0,1])\hc\lower.2pt\hbox{$D$}\supset M([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

69. abcde bde cde de de

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(1,2)\cup(2,3);\nq B\lw\{1\};\nq C\lw\{2\}\}\hc\hskip-1pt\begin{align}\sk{D}M([0,1])&\\\sk{E}F'([0,1])\end{align}\hc}\right.$

70. acde bcde cde cde e

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup\bigcup_{n=0}^\infty G_{2n+1}([1,2])\cup(2,3);\nq B\lw\bigcup_{n=0}^\infty G_{2n+2}([1,2]);\nq C\lw M([1,2]);\nq D\lw F'([1,2])\}\hc\lower.2pt\hbox{$E$}\supset F([0,1])\hc}\right.$

71. abcde b cde cde cde

    $[0,3)\ss{G([0,1])\cup(1,2)\cup(2,3)}{\{2\}}{\bigcup_{n=0}^\infty M_{2n}([0,1])}{\bigcup_{n=0}^\infty M_{2n+1}([0,1])}{F'([0,1])}$

72. ae bcde bcde bcde e

    $[0,2)\ss{(0,1)}{W_1((1,2))}{W_2((1,2))}{W_3((1,2))}{\{0,1\}}$

73. abcde bcde cde de e

    $[0,1)\ss{(0,1)\sm(B\cup C\cup D)}{h^-_C((0,1))\sm D}{h^-_D((0,1))}{h^-_{\{0\}}([0,1))}{\{0\}}$

74. abcde abcde ce de e

    $[0,1)\ss{W_1((0,1))\sm(C\cup D)}{W_2((0,1))\sm(C\cup D)}{h^-_{\{0\}}([0,1))\sm D}{h^+_{\{1\}}((0,1])}{\{0\}}$

75. abcde abcde c de de

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw W_1((0,1));\nq B\lw W_2((0,1))\}\hc\lower.2pt\hbox{$C$}\supset M_0([0,1])\hc\lower.2pt\hbox{$D$}\supset\bigcup_{n=1}^\infty M_n([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

76. abcde bcde bcde d e

    $[0,4)\left\{\matrix{\left(H^+([0,1))\cup H^+([2,3))\right)\{\nq A\lw G([0,1])\cup(1,2);\nq B\lw M([0,1]);\nq C\lw F'([0,1])\}\hc\begin{align}\sk{A}(1,2)\cup(3,4)&\\\sk{D}\{1\}&\\\sk{E}\{3\}\end{align}\hc}\right.$

77. abcde abcde cde d e

    $[0,2)\ss{W_1((0,1)\cup(1,2))\sm C}{W_2((0,1)\cup(1,2))\sm C}{h^-_{\{0\}}([0,1))\cup h^-_{\{1\}}([1,2))}{\{0\}}{\{1\}}$

78. abcde bcde cde de de

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(1,2)\sm B;\nq B\lw h^-_{\{1\}}([1,2));\nq C\lw\{1\}\}\hc\lower.2pt\hbox{$D$}\supset M([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

79. abcde bcde cde cde e

    $[0,2)\left\{\matrix{H^+([1,2))\{\nq G([0,1])\{\nqq A\lw(0,1)\cup(1,2);\nqq B\lw\{1\}\};\nq A\lw(1,2);\nq C\lw M([0,1]);\nq D\lw F'([0,1])\}\hc\lower.2pt\hbox{$A$}\supset(0,1)\hc\lower.2pt\hbox{$E$}\supset\{0\}\hc}\right.$

80. abcde bcde bcde de e

    $[0,2)\left\{\matrix{H^+([1,2))\{\nq A\lw G([0,1])\cup(1,2);\nq B\lw\bigcup_{n=1}^\infty M_n([0,1]);\nq C\lw F'([0,1]);\nq D\lw M_0([0,1])\}\hc\lower.2pt\hbox{$A$}\supset(0,1)\hc\lower.2pt\hbox{$E$}\supset\{0\}\hc}\right.$

81. abcde abcde cde de e

    $[0,1)\ss{W_1((0,1))\sm(C\cup D)}{W_2((0,1))\sm(C\cup D)}{h^-_D((0,1))}{h^-_{\{0\}}([0,1))}{\{0\}}$

82. acde bcde cde cde cde

    $[0,1]\ss{\bigcup_{n=0}^\infty G_{2n+1}([0,1])}{\bigcup_{n=0}^\infty G_{2n+2}([0,1])}{\bigcup_{n=0}^\infty M_{2n}([0,1])}{\bigcup_{n=0}^\infty M_{2n+1}([0,1])}{F'([0,1])}$

83. abcde bcde bcde de de

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw (0,1)\cup G([1,2])\cup(2,3);\nq B\lw M([1,2]);\nq C\lw F'([1,2])\}\hc\lower.2pt\hbox{$D$}\supset M([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

84. abcde abcde cde de de

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw W_1((0,1)\cup(1,2));\nq B\lw W_2((0,1)\cup(1,2));\nq C\lw\{1\}\}\hc\lower.2pt\hbox{$D$}\supset M([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

85. abcde abcde cde cde e

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw W_1((0,1)\cup G([1,2])\cup(2,3));\nq B\lw W_2((0,1)\cup G([1,2])\cup(2,3));\nq C\lw M([1,2]);\nq D\lw F'([1,2])\}\hc\lower.2pt\hbox{$D$}\supset M([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

86. abcde abcde abcde d e

    $[0,2)\ss{W_1((0,1)\cup(1,2))}{W_2((0,1)\cup(1,2))}{W_3((0,1)\cup(1,2))}{\{0\}}{\{1\}}$

87. abcde bcde cde cde cde

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw(0,1)\cup(1,2);\nq B\lw\{1\}\}\hc\lower.2pt\hbox{$C$}\supset\bigcup_{n=0}^\infty M_{2n}([0,1])\hc\lower.2pt\hbox{$D$}\supset\bigcup_{n=0}^\infty M_{2n+1}([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

88. abcde bcde bcde bcde e

    $[0,1]\ss{G([0,1])}{\bigcup_{n=0}^\infty M_{2n+1}([0,1])}{\bigcup_{n=0}^\infty M_{2n+2}([0,1])}{F'([0,1])}{M_0([0,1])}$

89. abcde abcde abcde de e

    $[0,1)\ss{W_1((0,1))\sm D}{W_2((0,1))\sm D}{W_3((0,1))\sm D}{h^-_{\{0\}}([0,1))}{\{0\}}$

90. abcde abcde cde cde cde

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw W_1((0,1));\nq B\lw W_2((0,1))\}\hc\lower.2pt\hbox{$C$}\supset\bigcup_{n=0}^\infty M_{2n}([0,1])\hc\lower.2pt\hbox{$D$}\supset\bigcup_{n=0}^\infty M_{2n+1}([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

91. abcde abcde abcde de de

    $[0,1]\left\{\matrix{G([0,1])\{\nq A\lw W_1((0,1));\nq B\lw W_2((0,1));\nq C\lw W_3((0,1))\}\hc\lower.2pt\hbox{$D$}\supset M([0,1])\hc\lower.2pt\hbox{$E$}\supset F'([0,1])\hc}\right.$

92. abcde bcde bcde bcde bcde

    $[0,1]\ss{G([0,1])}{\bigcup_{n=0}^\infty M_{3n}([0,1])}{\bigcup_{n=0}^\infty M_{3n+1}([0,1])}{\bigcup_{n=0}^\infty M_{3n+2}([0,1])}{F'([0,1])}$

93. abcde abcde abcde abcde e

    $\sp{W_1(\R\sm\{0\})}{W_2(\R\sm\{0\})}{W_3(\R\sm\{0\})}{W_4(\R\sm\{0\})}{\{0\}}$

94. abcde abcde abcde abcde abcde

    $\sp{W_1(\R)}{W_2(\R)}{W_3(\R)}{W_4(\R)}{W_5(\R)}$