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Example of Set X which is not well ordered such that $Xtimes [0,1)$ with dictionary ordered is not Linear Continuum

Is $[0,1)times[0,1]$ a linear continuum?A connected linearly ordered set is a linear continuumExercise 24.6 Munkres: $X times [0,1)$ is a linear continuumExample 2, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: How to show this set to be a linear continuum?can you show example for this?Let $X=[0,1]times [0,1]$. We denote the set $ X $ with the subspace topology of $mathbbRtimes mathbbR$ …Which are linear continua?Example 2, Sec. 24, in Munkres' TOPOLOGY, 2nd ed: If $X$ is a well ordered set, then $X times [0, 1)$ is a linear continuumProb. 4, Sec. 24, in Munkres' TOPOLOGY, 2nd ed: If an ordered set in the order topology is connected, then it is a linear continuumProb. 6, Sec. 24, in Munkres' TOPOLOGY, 2nd ed: For a well-ordered set $X$, $Xtimes[0,1)$ in the dictionary order is a linear continuum

0

$begingroup$

Example of Set X which is not well ordered such that $Xtimes [0,1)$ with dictionary ordered is not Linear Continuum

I had proved following theorem

Set X which is not well ordered then $Xtimes [0,1)$ with dictionary ordered is Linear Continuum.

Linear Continuum have the property of LUB and for any 2 point of its there is point between them.

I wanted to know is there converse true or not?

I found all example till that are well ordered

But I was suspicous about converse .

Please give me hint to construct a counterexample

Any Help will be appreciated

general-topology connectedness

share|cite|improve this question

asked Mar 21 at 4:14

SRJSRJ

1,9151620

$endgroup$

add a comment | 

0

$begingroup$

Example of Set X which is not well ordered such that $Xtimes [0,1)$ with dictionary ordered is not Linear Continuum

I had proved following theorem

Set X which is not well ordered then $Xtimes [0,1)$ with dictionary ordered is Linear Continuum.

Linear Continuum have the property of LUB and for any 2 point of its there is point between them.

I wanted to know is there converse true or not?

I found all example till that are well ordered

But I was suspicous about converse .

Please give me hint to construct a counterexample

Any Help will be appreciated

general-topology connectedness

share|cite|improve this question

asked Mar 21 at 4:14

SRJSRJ

1,9151620

$endgroup$

add a comment | 

$begingroup$

Example of Set X which is not well ordered such that $Xtimes [0,1)$ with dictionary ordered is not Linear Continuum

I had proved following theorem

Set X which is not well ordered then $Xtimes [0,1)$ with dictionary ordered is Linear Continuum.

Linear Continuum have the property of LUB and for any 2 point of its there is point between them.

I wanted to know is there converse true or not?

I found all example till that are well ordered

But I was suspicous about converse .

Please give me hint to construct a counterexample

Any Help will be appreciated

general-topology connectedness

share|cite|improve this question

asked Mar 21 at 4:14

SRJSRJ

1,9151620

$endgroup$

share|cite|improve this question

asked Mar 21 at 4:14

SRJSRJ

1,9151620

share|cite|improve this question

asked Mar 21 at 4:14

SRJSRJ

1,9151620

asked Mar 21 at 4:14

SRJSRJ

1,9151620

asked Mar 21 at 4:14

SRJSRJ

1,9151620

1 Answer
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If $X times_textrmlex [0,1)$ is a continuum, $X$ must have the property that a non-maximal element has a successor $x^+$: What else can the supremum of $x times [0,1)$ else be but $(x^+,0)$, where $(x,x^+)subseteq X$ is empty? This property is implied by being a well-order, but it might be almost equivalent to what you want. $X=mathbbZ$ is an example: note that $mathbbZ times_textrmlex [0,1)$ is homeomorphic to $mathbbR$, a continuum (via $x to (lfloor xrfloor, x - lfloor xrfloor )$, e.g., while $mathbbZ$ is not well-ordered.

share|cite|improve this answer

edited Mar 21 at 16:22

answered Mar 21 at 5:11

Henno BrandsmaHenno Brandsma

115k349125

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add a comment | 

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1

$begingroup$

If $X times_textrmlex [0,1)$ is a continuum, $X$ must have the property that a non-maximal element has a successor $x^+$: What else can the supremum of $x times [0,1)$ else be but $(x^+,0)$, where $(x,x^+)subseteq X$ is empty? This property is implied by being a well-order, but it might be almost equivalent to what you want. $X=mathbbZ$ is an example: note that $mathbbZ times_textrmlex [0,1)$ is homeomorphic to $mathbbR$, a continuum (via $x to (lfloor xrfloor, x - lfloor xrfloor )$, e.g., while $mathbbZ$ is not well-ordered.

share|cite|improve this answer

edited Mar 21 at 16:22

answered Mar 21 at 5:11

Henno BrandsmaHenno Brandsma

115k349125

$endgroup$

add a comment | 

1

$begingroup$

If $X times_textrmlex [0,1)$ is a continuum, $X$ must have the property that a non-maximal element has a successor $x^+$: What else can the supremum of $x times [0,1)$ else be but $(x^+,0)$, where $(x,x^+)subseteq X$ is empty? This property is implied by being a well-order, but it might be almost equivalent to what you want. $X=mathbbZ$ is an example: note that $mathbbZ times_textrmlex [0,1)$ is homeomorphic to $mathbbR$, a continuum (via $x to (lfloor xrfloor, x - lfloor xrfloor )$, e.g., while $mathbbZ$ is not well-ordered.

share|cite|improve this answer

edited Mar 21 at 16:22

answered Mar 21 at 5:11

Henno BrandsmaHenno Brandsma

115k349125

$endgroup$

add a comment | 

$begingroup$

If $X times_textrmlex [0,1)$ is a continuum, $X$ must have the property that a non-maximal element has a successor $x^+$: What else can the supremum of $x times [0,1)$ else be but $(x^+,0)$, where $(x,x^+)subseteq X$ is empty? This property is implied by being a well-order, but it might be almost equivalent to what you want. $X=mathbbZ$ is an example: note that $mathbbZ times_textrmlex [0,1)$ is homeomorphic to $mathbbR$, a continuum (via $x to (lfloor xrfloor, x - lfloor xrfloor )$, e.g., while $mathbbZ$ is not well-ordered.

share|cite|improve this answer

edited Mar 21 at 16:22

answered Mar 21 at 5:11

Henno BrandsmaHenno Brandsma

115k349125

$endgroup$

share|cite|improve this answer

edited Mar 21 at 16:22

answered Mar 21 at 5:11

Henno BrandsmaHenno Brandsma

115k349125

answered Mar 21 at 5:11

Henno BrandsmaHenno Brandsma

115k349125

answered Mar 21 at 5:11

Henno BrandsmaHenno Brandsma

115k349125