Continuity of two variables in toplogical space .
$begingroup$
Let $T$ is subset of real numbers .Let $k$ is any number which is not in $T$ .
Define $$kT ={ kt : t in T}. $$
Let $W$ is toplogy(not usual) on $T$ , then can you prove that :
$$H={ kU : U in W }$$ is topology for $kT$ .
And is the function $f:kTtimes kTto kT$ continuous with respect to $H$ if it is defined as
$$f (kt,ks)=k (t+s ),$$ for $t ,s in T$?
Note that it is given : for every $s,t in T $, $s+t$ belongs to $T .$
real-analysis general-topology continuity metric-spaces metrizability
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,597917
asked Dec 27 '18 at 8:12
GuruGuru
63
$endgroup$
add a comment |
$begingroup$
Let $T$ is subset of real numbers .Let $k$ is any number which is not in $T$ .
Define $$kT ={ kt : t in T}. $$
Let $W$ is toplogy(not usual) on $T$ , then can you prove that :
$$H={ kU : U in W }$$ is topology for $kT$ .
And is the function $f:kTtimes kTto kT$ continuous with respect to $H$ if it is defined as
$$f (kt,ks)=k (t+s ),$$ for $t ,s in T$?
Note that it is given : for every $s,t in T $, $s+t$ belongs to $T .$
real-analysis general-topology continuity metric-spaces metrizability
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,597917
asked Dec 27 '18 at 8:12
GuruGuru
63
$endgroup$
$begingroup$
The second line is unclear. Can you rephrase it?
$endgroup$
– caffeinemachine
Dec 27 '18 at 8:16
$begingroup$
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
$endgroup$
– Guru
Dec 27 '18 at 9:41
$begingroup$
@Guru I had misread the second part. I have posted an answer now.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 10:04
add a comment |
$begingroup$
Let $T$ is subset of real numbers .Let $k$ is any number which is not in $T$ .
Define $$kT ={ kt : t in T}. $$
Let $W$ is toplogy(not usual) on $T$ , then can you prove that :
$$H={ kU : U in W }$$ is topology for $kT$ .
And is the function $f:kTtimes kTto kT$ continuous with respect to $H$ if it is defined as
$$f (kt,ks)=k (t+s ),$$ for $t ,s in T$?
Note that it is given : for every $s,t in T $, $s+t$ belongs to $T .$
real-analysis general-topology continuity metric-spaces metrizability
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,597917
asked Dec 27 '18 at 8:12
GuruGuru
63
$endgroup$
Let $T$ is subset of real numbers .Let $k$ is any number which is not in $T$ .
Define $$kT ={ kt : t in T}. $$
Let $W$ is toplogy(not usual) on $T$ , then can you prove that :
$$H={ kU : U in W }$$ is topology for $kT$ .
And is the function $f:kTtimes kTto kT$ continuous with respect to $H$ if it is defined as
$$f (kt,ks)=k (t+s ),$$ for $t ,s in T$?
Note that it is given : for every $s,t in T $, $s+t$ belongs to $T .$
real-analysis general-topology continuity metric-spaces metrizability
real-analysis general-topology continuity metric-spaces metrizability
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,597917
asked Dec 27 '18 at 8:12
GuruGuru
63
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,597917
asked Dec 27 '18 at 8:12
GuruGuru
63
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,597917
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,597917
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,597917
6,597917
asked Dec 27 '18 at 8:12
GuruGuru
63
asked Dec 27 '18 at 8:12
GuruGuru
63
asked Dec 27 '18 at 8:12
GuruGuru
63
63
$begingroup$
The second line is unclear. Can you rephrase it?
$endgroup$
– caffeinemachine
Dec 27 '18 at 8:16
$begingroup$
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
$endgroup$
– Guru
Dec 27 '18 at 9:41
$begingroup$
@Guru I had misread the second part. I have posted an answer now.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 10:04
add a comment |
$begingroup$
The second line is unclear. Can you rephrase it?
$endgroup$
– caffeinemachine
Dec 27 '18 at 8:16
$begingroup$
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
$endgroup$
– Guru
Dec 27 '18 at 9:41
$begingroup$
@Guru I had misread the second part. I have posted an answer now.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 10:04
$begingroup$
The second line is unclear. Can you rephrase it?
$endgroup$
– caffeinemachine
Dec 27 '18 at 8:16
$begingroup$
The second line is unclear. Can you rephrase it?
$endgroup$
– caffeinemachine
Dec 27 '18 at 8:16
$begingroup$
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
$endgroup$
– Guru
Dec 27 '18 at 9:41
$begingroup$
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
$endgroup$
– Guru
Dec 27 '18 at 9:41
$begingroup$
@Guru I had misread the second part. I have posted an answer now.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 10:04
$begingroup$
@Guru I had misread the second part. I have posted an answer now.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 10:04
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Thew first part follows from the relations $k(cup U_i)=cup kU_i$, $k(U cap V)=kUcap kV$, $kemptyset =kemptyset$. For the second part, taking $T=(1,infty)$ and $k=1$ you are asking if $(t,s) to t+s$ is continuous for ANY topology on $(1,infty)$. This is not true.
answered Dec 27 '18 at 9:56
Kavi Rama MurthyKavi Rama Murthy
62.5k42262
$endgroup$
$begingroup$
Kavi Rama murthy thanks for your reply .But i am asking how can we check continuity of above function with respect to topology H? Not on arbitrary toplogy .Thanks onece again.
$endgroup$
– Guru
Dec 27 '18 at 11:06
$begingroup$
@Guru When $k=1$ $kT=T$ and the topology on $kT$ is same as the topology on $T$. The topology yon $T$ is not specified. It can be any topology. So the map $(t,s) to t+s$ need not be continuous.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 11:45
$begingroup$
Ok... thanks a lot sir , kavi Rama murthy .one last qeustion " which book is the best to understand the concept of " toplogical fields " , homeomorphism between toplogical fields and homeomorphism on field structure together with isomorphisim for beggner"?
$endgroup$
– Guru
Dec 27 '18 at 14:58
add a comment |
Your Answer
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1 Answer
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1 Answer
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oldest
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oldest
votes
$begingroup$
Thew first part follows from the relations $k(cup U_i)=cup kU_i$, $k(U cap V)=kUcap kV$, $kemptyset =kemptyset$. For the second part, taking $T=(1,infty)$ and $k=1$ you are asking if $(t,s) to t+s$ is continuous for ANY topology on $(1,infty)$. This is not true.
answered Dec 27 '18 at 9:56
Kavi Rama MurthyKavi Rama Murthy
62.5k42262
$endgroup$
$begingroup$
Kavi Rama murthy thanks for your reply .But i am asking how can we check continuity of above function with respect to topology H? Not on arbitrary toplogy .Thanks onece again.
$endgroup$
– Guru
Dec 27 '18 at 11:06
$begingroup$
@Guru When $k=1$ $kT=T$ and the topology on $kT$ is same as the topology on $T$. The topology yon $T$ is not specified. It can be any topology. So the map $(t,s) to t+s$ need not be continuous.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 11:45
$begingroup$
Ok... thanks a lot sir , kavi Rama murthy .one last qeustion " which book is the best to understand the concept of " toplogical fields " , homeomorphism between toplogical fields and homeomorphism on field structure together with isomorphisim for beggner"?
$endgroup$
– Guru
Dec 27 '18 at 14:58
add a comment |
$begingroup$
Thew first part follows from the relations $k(cup U_i)=cup kU_i$, $k(U cap V)=kUcap kV$, $kemptyset =kemptyset$. For the second part, taking $T=(1,infty)$ and $k=1$ you are asking if $(t,s) to t+s$ is continuous for ANY topology on $(1,infty)$. This is not true.
answered Dec 27 '18 at 9:56
Kavi Rama MurthyKavi Rama Murthy
62.5k42262
$endgroup$
$begingroup$
Kavi Rama murthy thanks for your reply .But i am asking how can we check continuity of above function with respect to topology H? Not on arbitrary toplogy .Thanks onece again.
$endgroup$
– Guru
Dec 27 '18 at 11:06
$begingroup$
@Guru When $k=1$ $kT=T$ and the topology on $kT$ is same as the topology on $T$. The topology yon $T$ is not specified. It can be any topology. So the map $(t,s) to t+s$ need not be continuous.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 11:45
$begingroup$
Ok... thanks a lot sir , kavi Rama murthy .one last qeustion " which book is the best to understand the concept of " toplogical fields " , homeomorphism between toplogical fields and homeomorphism on field structure together with isomorphisim for beggner"?
$endgroup$
– Guru
Dec 27 '18 at 14:58
add a comment |
$begingroup$
Thew first part follows from the relations $k(cup U_i)=cup kU_i$, $k(U cap V)=kUcap kV$, $kemptyset =kemptyset$. For the second part, taking $T=(1,infty)$ and $k=1$ you are asking if $(t,s) to t+s$ is continuous for ANY topology on $(1,infty)$. This is not true.
answered Dec 27 '18 at 9:56
Kavi Rama MurthyKavi Rama Murthy
62.5k42262
$endgroup$
Thew first part follows from the relations $k(cup U_i)=cup kU_i$, $k(U cap V)=kUcap kV$, $kemptyset =kemptyset$. For the second part, taking $T=(1,infty)$ and $k=1$ you are asking if $(t,s) to t+s$ is continuous for ANY topology on $(1,infty)$. This is not true.
answered Dec 27 '18 at 9:56
Kavi Rama MurthyKavi Rama Murthy
62.5k42262
answered Dec 27 '18 at 9:56
Kavi Rama MurthyKavi Rama Murthy
62.5k42262
answered Dec 27 '18 at 9:56
Kavi Rama MurthyKavi Rama Murthy
62.5k42262
answered Dec 27 '18 at 9:56
Kavi Rama MurthyKavi Rama Murthy
62.5k42262
62.5k42262
$begingroup$
Kavi Rama murthy thanks for your reply .But i am asking how can we check continuity of above function with respect to topology H? Not on arbitrary toplogy .Thanks onece again.
$endgroup$
– Guru
Dec 27 '18 at 11:06
$begingroup$
@Guru When $k=1$ $kT=T$ and the topology on $kT$ is same as the topology on $T$. The topology yon $T$ is not specified. It can be any topology. So the map $(t,s) to t+s$ need not be continuous.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 11:45
$begingroup$
Ok... thanks a lot sir , kavi Rama murthy .one last qeustion " which book is the best to understand the concept of " toplogical fields " , homeomorphism between toplogical fields and homeomorphism on field structure together with isomorphisim for beggner"?
$endgroup$
– Guru
Dec 27 '18 at 14:58
add a comment |
$begingroup$
Kavi Rama murthy thanks for your reply .But i am asking how can we check continuity of above function with respect to topology H? Not on arbitrary toplogy .Thanks onece again.
$endgroup$
– Guru
Dec 27 '18 at 11:06
$begingroup$
@Guru When $k=1$ $kT=T$ and the topology on $kT$ is same as the topology on $T$. The topology yon $T$ is not specified. It can be any topology. So the map $(t,s) to t+s$ need not be continuous.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 11:45
$begingroup$
Ok... thanks a lot sir , kavi Rama murthy .one last qeustion " which book is the best to understand the concept of " toplogical fields " , homeomorphism between toplogical fields and homeomorphism on field structure together with isomorphisim for beggner"?
$endgroup$
– Guru
Dec 27 '18 at 14:58
$begingroup$
Kavi Rama murthy thanks for your reply .But i am asking how can we check continuity of above function with respect to topology H? Not on arbitrary toplogy .Thanks onece again.
$endgroup$
– Guru
Dec 27 '18 at 11:06
$begingroup$
Kavi Rama murthy thanks for your reply .But i am asking how can we check continuity of above function with respect to topology H? Not on arbitrary toplogy .Thanks onece again.
$endgroup$
– Guru
Dec 27 '18 at 11:06
$begingroup$
@Guru When $k=1$ $kT=T$ and the topology on $kT$ is same as the topology on $T$. The topology yon $T$ is not specified. It can be any topology. So the map $(t,s) to t+s$ need not be continuous.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 11:45
$begingroup$
@Guru When $k=1$ $kT=T$ and the topology on $kT$ is same as the topology on $T$. The topology yon $T$ is not specified. It can be any topology. So the map $(t,s) to t+s$ need not be continuous.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 11:45
$begingroup$
Ok... thanks a lot sir , kavi Rama murthy .one last qeustion " which book is the best to understand the concept of " toplogical fields " , homeomorphism between toplogical fields and homeomorphism on field structure together with isomorphisim for beggner"?
$endgroup$
– Guru
Dec 27 '18 at 14:58
$begingroup$
Ok... thanks a lot sir , kavi Rama murthy .one last qeustion " which book is the best to understand the concept of " toplogical fields " , homeomorphism between toplogical fields and homeomorphism on field structure together with isomorphisim for beggner"?
$endgroup$
– Guru
Dec 27 '18 at 14:58
add a comment |
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$begingroup$
The second line is unclear. Can you rephrase it?
$endgroup$
– caffeinemachine
Dec 27 '18 at 8:16
$begingroup$
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
$endgroup$
– Guru
Dec 27 '18 at 9:41
$begingroup$
@Guru I had misread the second part. I have posted an answer now.
$endgroup$
– Kavi Rama Murthy
Dec 27 '18 at 10:04