$(-sqrt 2, sqrt 2) cap mathbb Q$ in order topology. What will be it's bases?
$begingroup$
$(-sqrt 2, sqrt 2) cap mathbb Q$ in order topology. What will be its bases?
My try: [ $(a,b)$ where $a,b in (-sqrt2, sqrt2) cap mathbb Q$ and $(-infty , b)$ , $(a, infty)$ {because supremum and infimum of $(-sqrt 2, sqrt 2) cap mathbb Q$ do not exist}]
Can anyone please correct me if I go wrong anywhere?
general-topology
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add a comment |
$begingroup$
$(-sqrt 2, sqrt 2) cap mathbb Q$ in order topology. What will be its bases?
My try: [ $(a,b)$ where $a,b in (-sqrt2, sqrt2) cap mathbb Q$ and $(-infty , b)$ , $(a, infty)$ {because supremum and infimum of $(-sqrt 2, sqrt 2) cap mathbb Q$ do not exist}]
Can anyone please correct me if I go wrong anywhere?
general-topology
$endgroup$
add a comment |
$begingroup$
$(-sqrt 2, sqrt 2) cap mathbb Q$ in order topology. What will be its bases?
My try: [ $(a,b)$ where $a,b in (-sqrt2, sqrt2) cap mathbb Q$ and $(-infty , b)$ , $(a, infty)$ {because supremum and infimum of $(-sqrt 2, sqrt 2) cap mathbb Q$ do not exist}]
Can anyone please correct me if I go wrong anywhere?
general-topology
$endgroup$
$(-sqrt 2, sqrt 2) cap mathbb Q$ in order topology. What will be its bases?
My try: [ $(a,b)$ where $a,b in (-sqrt2, sqrt2) cap mathbb Q$ and $(-infty , b)$ , $(a, infty)$ {because supremum and infimum of $(-sqrt 2, sqrt 2) cap mathbb Q$ do not exist}]
Can anyone please correct me if I go wrong anywhere?
general-topology
general-topology
edited Jul 28 '18 at 6:48
celtschk
29.9k755101
29.9k755101
asked Jul 28 '18 at 1:54
cmicmi
1,109212
1,109212
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add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Looks like you mostly have the right idea! I will say that you're expressing it a bit awkwardly, though.
I would put it this way:
One basis for the order topology on $X:=left(-sqrt{2},sqrt{2}right)capBbb Q$ is $$bigl{(a,b)cap Xmid a,bin Xbigr}cupbigl{(-infty,b)cap Xmid bin Xbigr}cupbigl{(a,infty)cap Xmid ain Xbigr}.$$
At that point, you must justify that this is a basis for the given topology. It sounds like you have some results you can use to help you accomplish that.
Let me know if you have any trouble justifying the claims, or seeing the many differences between your version and mine.
Added: As an exercise, you should try to prove that $bigl{(a,b)cap Xmid a,bin Xbigr}$ is a basis for the topology on $X,$ all on its own. This is the main reason why I emphasized the word "One" in my rewording above. Another is that the entire topology also comprises a basis (though it's much less straightforward to describe).
$endgroup$
1
$begingroup$
It probably deserves mention that the last two terms in that union can be omitted -- the first one is itself a basis.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:17
$begingroup$
@Henning: Fair point. I thought I'd minimize the "burden of proof" (pun intended) on the OP.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:18
1
$begingroup$
Also, I suspect the OP may be working in a context where, say, $(a,infty)$ doesn't necessarily mean a real half-interval, but an interval of whatever ordered set we're working in -- here, $X$. If that is the case, you explicit intersections with $X$ are not needed.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:21
$begingroup$
@Henning: I wondered about that, too. I thought it better to be redundant, just in case, rather than assume too much.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:23
add a comment |
$begingroup$
Let r = $sqrt 2.$ A simple base is
{ (a,b) $cap$ Q : a,b in (-r,r) }.
Exercise. Show a subbase is
{ (-r,a) $cap$ Q, (a,r) $cap$ Q : a in (-r,r) }.
In both definitions a,b can be futher restricted to rationals.
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
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active
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votes
$begingroup$
Looks like you mostly have the right idea! I will say that you're expressing it a bit awkwardly, though.
I would put it this way:
One basis for the order topology on $X:=left(-sqrt{2},sqrt{2}right)capBbb Q$ is $$bigl{(a,b)cap Xmid a,bin Xbigr}cupbigl{(-infty,b)cap Xmid bin Xbigr}cupbigl{(a,infty)cap Xmid ain Xbigr}.$$
At that point, you must justify that this is a basis for the given topology. It sounds like you have some results you can use to help you accomplish that.
Let me know if you have any trouble justifying the claims, or seeing the many differences between your version and mine.
Added: As an exercise, you should try to prove that $bigl{(a,b)cap Xmid a,bin Xbigr}$ is a basis for the topology on $X,$ all on its own. This is the main reason why I emphasized the word "One" in my rewording above. Another is that the entire topology also comprises a basis (though it's much less straightforward to describe).
$endgroup$
1
$begingroup$
It probably deserves mention that the last two terms in that union can be omitted -- the first one is itself a basis.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:17
$begingroup$
@Henning: Fair point. I thought I'd minimize the "burden of proof" (pun intended) on the OP.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:18
1
$begingroup$
Also, I suspect the OP may be working in a context where, say, $(a,infty)$ doesn't necessarily mean a real half-interval, but an interval of whatever ordered set we're working in -- here, $X$. If that is the case, you explicit intersections with $X$ are not needed.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:21
$begingroup$
@Henning: I wondered about that, too. I thought it better to be redundant, just in case, rather than assume too much.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:23
add a comment |
$begingroup$
Looks like you mostly have the right idea! I will say that you're expressing it a bit awkwardly, though.
I would put it this way:
One basis for the order topology on $X:=left(-sqrt{2},sqrt{2}right)capBbb Q$ is $$bigl{(a,b)cap Xmid a,bin Xbigr}cupbigl{(-infty,b)cap Xmid bin Xbigr}cupbigl{(a,infty)cap Xmid ain Xbigr}.$$
At that point, you must justify that this is a basis for the given topology. It sounds like you have some results you can use to help you accomplish that.
Let me know if you have any trouble justifying the claims, or seeing the many differences between your version and mine.
Added: As an exercise, you should try to prove that $bigl{(a,b)cap Xmid a,bin Xbigr}$ is a basis for the topology on $X,$ all on its own. This is the main reason why I emphasized the word "One" in my rewording above. Another is that the entire topology also comprises a basis (though it's much less straightforward to describe).
$endgroup$
1
$begingroup$
It probably deserves mention that the last two terms in that union can be omitted -- the first one is itself a basis.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:17
$begingroup$
@Henning: Fair point. I thought I'd minimize the "burden of proof" (pun intended) on the OP.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:18
1
$begingroup$
Also, I suspect the OP may be working in a context where, say, $(a,infty)$ doesn't necessarily mean a real half-interval, but an interval of whatever ordered set we're working in -- here, $X$. If that is the case, you explicit intersections with $X$ are not needed.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:21
$begingroup$
@Henning: I wondered about that, too. I thought it better to be redundant, just in case, rather than assume too much.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:23
add a comment |
$begingroup$
Looks like you mostly have the right idea! I will say that you're expressing it a bit awkwardly, though.
I would put it this way:
One basis for the order topology on $X:=left(-sqrt{2},sqrt{2}right)capBbb Q$ is $$bigl{(a,b)cap Xmid a,bin Xbigr}cupbigl{(-infty,b)cap Xmid bin Xbigr}cupbigl{(a,infty)cap Xmid ain Xbigr}.$$
At that point, you must justify that this is a basis for the given topology. It sounds like you have some results you can use to help you accomplish that.
Let me know if you have any trouble justifying the claims, or seeing the many differences between your version and mine.
Added: As an exercise, you should try to prove that $bigl{(a,b)cap Xmid a,bin Xbigr}$ is a basis for the topology on $X,$ all on its own. This is the main reason why I emphasized the word "One" in my rewording above. Another is that the entire topology also comprises a basis (though it's much less straightforward to describe).
$endgroup$
Looks like you mostly have the right idea! I will say that you're expressing it a bit awkwardly, though.
I would put it this way:
One basis for the order topology on $X:=left(-sqrt{2},sqrt{2}right)capBbb Q$ is $$bigl{(a,b)cap Xmid a,bin Xbigr}cupbigl{(-infty,b)cap Xmid bin Xbigr}cupbigl{(a,infty)cap Xmid ain Xbigr}.$$
At that point, you must justify that this is a basis for the given topology. It sounds like you have some results you can use to help you accomplish that.
Let me know if you have any trouble justifying the claims, or seeing the many differences between your version and mine.
Added: As an exercise, you should try to prove that $bigl{(a,b)cap Xmid a,bin Xbigr}$ is a basis for the topology on $X,$ all on its own. This is the main reason why I emphasized the word "One" in my rewording above. Another is that the entire topology also comprises a basis (though it's much less straightforward to describe).
edited Jan 6 at 19:11
answered Jul 28 '18 at 2:16
Cameron BuieCameron Buie
85.1k771155
85.1k771155
1
$begingroup$
It probably deserves mention that the last two terms in that union can be omitted -- the first one is itself a basis.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:17
$begingroup$
@Henning: Fair point. I thought I'd minimize the "burden of proof" (pun intended) on the OP.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:18
1
$begingroup$
Also, I suspect the OP may be working in a context where, say, $(a,infty)$ doesn't necessarily mean a real half-interval, but an interval of whatever ordered set we're working in -- here, $X$. If that is the case, you explicit intersections with $X$ are not needed.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:21
$begingroup$
@Henning: I wondered about that, too. I thought it better to be redundant, just in case, rather than assume too much.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:23
add a comment |
1
$begingroup$
It probably deserves mention that the last two terms in that union can be omitted -- the first one is itself a basis.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:17
$begingroup$
@Henning: Fair point. I thought I'd minimize the "burden of proof" (pun intended) on the OP.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:18
1
$begingroup$
Also, I suspect the OP may be working in a context where, say, $(a,infty)$ doesn't necessarily mean a real half-interval, but an interval of whatever ordered set we're working in -- here, $X$. If that is the case, you explicit intersections with $X$ are not needed.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:21
$begingroup$
@Henning: I wondered about that, too. I thought it better to be redundant, just in case, rather than assume too much.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:23
1
1
$begingroup$
It probably deserves mention that the last two terms in that union can be omitted -- the first one is itself a basis.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:17
$begingroup$
It probably deserves mention that the last two terms in that union can be omitted -- the first one is itself a basis.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:17
$begingroup$
@Henning: Fair point. I thought I'd minimize the "burden of proof" (pun intended) on the OP.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:18
$begingroup$
@Henning: Fair point. I thought I'd minimize the "burden of proof" (pun intended) on the OP.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:18
1
1
$begingroup$
Also, I suspect the OP may be working in a context where, say, $(a,infty)$ doesn't necessarily mean a real half-interval, but an interval of whatever ordered set we're working in -- here, $X$. If that is the case, you explicit intersections with $X$ are not needed.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:21
$begingroup$
Also, I suspect the OP may be working in a context where, say, $(a,infty)$ doesn't necessarily mean a real half-interval, but an interval of whatever ordered set we're working in -- here, $X$. If that is the case, you explicit intersections with $X$ are not needed.
$endgroup$
– Henning Makholm
Jul 28 '18 at 2:21
$begingroup$
@Henning: I wondered about that, too. I thought it better to be redundant, just in case, rather than assume too much.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:23
$begingroup$
@Henning: I wondered about that, too. I thought it better to be redundant, just in case, rather than assume too much.
$endgroup$
– Cameron Buie
Jul 28 '18 at 2:23
add a comment |
$begingroup$
Let r = $sqrt 2.$ A simple base is
{ (a,b) $cap$ Q : a,b in (-r,r) }.
Exercise. Show a subbase is
{ (-r,a) $cap$ Q, (a,r) $cap$ Q : a in (-r,r) }.
In both definitions a,b can be futher restricted to rationals.
$endgroup$
add a comment |
$begingroup$
Let r = $sqrt 2.$ A simple base is
{ (a,b) $cap$ Q : a,b in (-r,r) }.
Exercise. Show a subbase is
{ (-r,a) $cap$ Q, (a,r) $cap$ Q : a in (-r,r) }.
In both definitions a,b can be futher restricted to rationals.
$endgroup$
add a comment |
$begingroup$
Let r = $sqrt 2.$ A simple base is
{ (a,b) $cap$ Q : a,b in (-r,r) }.
Exercise. Show a subbase is
{ (-r,a) $cap$ Q, (a,r) $cap$ Q : a in (-r,r) }.
In both definitions a,b can be futher restricted to rationals.
$endgroup$
Let r = $sqrt 2.$ A simple base is
{ (a,b) $cap$ Q : a,b in (-r,r) }.
Exercise. Show a subbase is
{ (-r,a) $cap$ Q, (a,r) $cap$ Q : a in (-r,r) }.
In both definitions a,b can be futher restricted to rationals.
answered Jul 28 '18 at 5:24
William ElliotWilliam Elliot
7,5472720
7,5472720
add a comment |
add a comment |
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