Prove that lower limit and upper limit topologies on R are homeomorphic
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I'm trying to prove that lower limit and upper limit topologies on R are homeomorphic.
However, it is clear that the identity is not an homeomorphism because, for example, $[0,1)$ is open in the lower limit topology, but not in the upper limit topology, and this is true for any half-open set; it is open in one, but not in the other, so the homeomorphism has to map such half open intervals to the half intervals (in the opposite sense), but I cannot think any such map; even after thinking on it for 2 weeks.
Of course, it has to also map open intervals to open intervals too.
general-topology
asked Nov 17 at 12:43
onurcanbektas
3,2641935
add a comment |
up vote
0
down vote
favorite
I'm trying to prove that lower limit and upper limit topologies on R are homeomorphic.
However, it is clear that the identity is not an homeomorphism because, for example, $[0,1)$ is open in the lower limit topology, but not in the upper limit topology, and this is true for any half-open set; it is open in one, but not in the other, so the homeomorphism has to map such half open intervals to the half intervals (in the opposite sense), but I cannot think any such map; even after thinking on it for 2 weeks.
Of course, it has to also map open intervals to open intervals too.
general-topology
asked Nov 17 at 12:43
onurcanbektas
3,2641935
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm trying to prove that lower limit and upper limit topologies on R are homeomorphic.
However, it is clear that the identity is not an homeomorphism because, for example, $[0,1)$ is open in the lower limit topology, but not in the upper limit topology, and this is true for any half-open set; it is open in one, but not in the other, so the homeomorphism has to map such half open intervals to the half intervals (in the opposite sense), but I cannot think any such map; even after thinking on it for 2 weeks.
Of course, it has to also map open intervals to open intervals too.
general-topology
asked Nov 17 at 12:43
onurcanbektas
3,2641935
I'm trying to prove that lower limit and upper limit topologies on R are homeomorphic.
However, it is clear that the identity is not an homeomorphism because, for example, $[0,1)$ is open in the lower limit topology, but not in the upper limit topology, and this is true for any half-open set; it is open in one, but not in the other, so the homeomorphism has to map such half open intervals to the half intervals (in the opposite sense), but I cannot think any such map; even after thinking on it for 2 weeks.
Of course, it has to also map open intervals to open intervals too.
general-topology
general-topology
asked Nov 17 at 12:43
onurcanbektas
3,2641935
asked Nov 17 at 12:43
onurcanbektas
3,2641935
asked Nov 17 at 12:43
onurcanbektas
3,2641935
asked Nov 17 at 12:43
onurcanbektas
3,2641935
asked Nov 17 at 12:43
onurcanbektas
3,2641935
3,2641935
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
$f(x) = -x$ is a homeomorphism, If $a < b$, $f^{-1}[(a,b]] = [-b, -a)$ etc.
answered Nov 17 at 12:45
Henno Brandsma
102k344108
Well, thanks a lot. Apparently, I forgot one of the two trivial maps; identity and the maps that send every element to its "inverse".
– onurcanbektas
Nov 17 at 12:48
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
$f(x) = -x$ is a homeomorphism, If $a < b$, $f^{-1}[(a,b]] = [-b, -a)$ etc.
answered Nov 17 at 12:45
Henno Brandsma
102k344108
Well, thanks a lot. Apparently, I forgot one of the two trivial maps; identity and the maps that send every element to its "inverse".
– onurcanbektas
Nov 17 at 12:48
add a comment |
up vote
2
down vote
accepted
$f(x) = -x$ is a homeomorphism, If $a < b$, $f^{-1}[(a,b]] = [-b, -a)$ etc.
answered Nov 17 at 12:45
Henno Brandsma
102k344108
Well, thanks a lot. Apparently, I forgot one of the two trivial maps; identity and the maps that send every element to its "inverse".
– onurcanbektas
Nov 17 at 12:48
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
$f(x) = -x$ is a homeomorphism, If $a < b$, $f^{-1}[(a,b]] = [-b, -a)$ etc.
answered Nov 17 at 12:45
Henno Brandsma
102k344108
$f(x) = -x$ is a homeomorphism, If $a < b$, $f^{-1}[(a,b]] = [-b, -a)$ etc.
answered Nov 17 at 12:45
Henno Brandsma
102k344108
answered Nov 17 at 12:45
Henno Brandsma
102k344108
answered Nov 17 at 12:45
Henno Brandsma
102k344108
answered Nov 17 at 12:45
Henno Brandsma
102k344108
102k344108
Well, thanks a lot. Apparently, I forgot one of the two trivial maps; identity and the maps that send every element to its "inverse".
– onurcanbektas
Nov 17 at 12:48
add a comment |
Well, thanks a lot. Apparently, I forgot one of the two trivial maps; identity and the maps that send every element to its "inverse".
– onurcanbektas
Nov 17 at 12:48
Well, thanks a lot. Apparently, I forgot one of the two trivial maps; identity and the maps that send every element to its "inverse".
– onurcanbektas
Nov 17 at 12:48
Well, thanks a lot. Apparently, I forgot one of the two trivial maps; identity and the maps that send every element to its "inverse".
– onurcanbektas
Nov 17 at 12:48
add a comment |
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