Continuity of two variables in toplogical space .
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
6,542917
asked Dec 27 '18 at 8:12
Guru
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add a comment |
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
6,542917
asked Dec 27 '18 at 8:12
Guru
63
New contributor
Guru is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
The second line is unclear. Can you rephrase it?
– caffeinemachine
Dec 27 '18 at 8:16
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
– Guru
Dec 27 '18 at 9:41
@Guru I had misread the second part. I have posted an answer now.
– Kavi Rama Murthy
Dec 27 '18 at 10:04
add a comment |
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
6,542917
asked Dec 27 '18 at 8:12
Guru
63
New contributor
Guru is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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
6,542917
asked Dec 27 '18 at 8:12
Guru
63
New contributor
Guru is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,542917
asked Dec 27 '18 at 8:12
Guru
63
New contributor
Guru is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,542917
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,542917
edited Dec 27 '18 at 8:50
MotylaNogaTomkaMazura
6,542917
6,542917
asked Dec 27 '18 at 8:12
Guru
63
New contributor
Guru is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 27 '18 at 8:12
Guru
63
asked Dec 27 '18 at 8:12
Guru
63
63
New contributor
Guru is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Guru is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Guru is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
The second line is unclear. Can you rephrase it?
– caffeinemachine
Dec 27 '18 at 8:16
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
– Guru
Dec 27 '18 at 9:41
@Guru I had misread the second part. I have posted an answer now.
– Kavi Rama Murthy
Dec 27 '18 at 10:04
add a comment |
The second line is unclear. Can you rephrase it?
– caffeinemachine
Dec 27 '18 at 8:16
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
– Guru
Dec 27 '18 at 9:41
@Guru I had misread the second part. I have posted an answer now.
– Kavi Rama Murthy
Dec 27 '18 at 10:04
The second line is unclear. Can you rephrase it?
– caffeinemachine
Dec 27 '18 at 8:16
The second line is unclear. Can you rephrase it?
– caffeinemachine
Dec 27 '18 at 8:16
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
– Guru
Dec 27 '18 at 9:41
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
– Guru
Dec 27 '18 at 9:41
@Guru I had misread the second part. I have posted an answer now.
– Kavi Rama Murthy
Dec 27 '18 at 10:04
@Guru I had misread the second part. I have posted an answer now.
– Kavi Rama Murthy
Dec 27 '18 at 10:04
add a comment |
1 Answer
1
active
oldest
votes
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 Murthy
50.5k31854
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.
– Guru
Dec 27 '18 at 11:06
@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.
– Kavi Rama Murthy
Dec 27 '18 at 11:45
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"?
– Guru
Dec 27 '18 at 14:58
add a comment |
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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 Murthy
50.5k31854
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.
– Guru
Dec 27 '18 at 11:06
@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.
– Kavi Rama Murthy
Dec 27 '18 at 11:45
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"?
– Guru
Dec 27 '18 at 14:58
add a comment |
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 Murthy
50.5k31854
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.
– Guru
Dec 27 '18 at 11:06
@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.
– Kavi Rama Murthy
Dec 27 '18 at 11:45
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"?
– Guru
Dec 27 '18 at 14:58
add a comment |
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 Murthy
50.5k31854
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 Murthy
50.5k31854
answered Dec 27 '18 at 9:56
Kavi Rama Murthy
50.5k31854
answered Dec 27 '18 at 9:56
Kavi Rama Murthy
50.5k31854
answered Dec 27 '18 at 9:56
Kavi Rama Murthy
50.5k31854
50.5k31854
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.
– Guru
Dec 27 '18 at 11:06
@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.
– Kavi Rama Murthy
Dec 27 '18 at 11:45
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"?
– Guru
Dec 27 '18 at 14:58
add a comment |
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.
– Guru
Dec 27 '18 at 11:06
@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.
– Kavi Rama Murthy
Dec 27 '18 at 11:45
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"?
– Guru
Dec 27 '18 at 14:58
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.
– Guru
Dec 27 '18 at 11:06
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.
– Guru
Dec 27 '18 at 11:06
@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.
– Kavi Rama Murthy
Dec 27 '18 at 11:45
@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.
– Kavi Rama Murthy
Dec 27 '18 at 11:45
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"?
– Guru
Dec 27 '18 at 14:58
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"?
– Guru
Dec 27 '18 at 14:58
add a comment |
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The second line is unclear. Can you rephrase it?
– caffeinemachine
Dec 27 '18 at 8:16
Kavi Rama Murthy can you explain it little more. How by definition it will hold?
– Guru
Dec 27 '18 at 9:41
@Guru I had misread the second part. I have posted an answer now.
– Kavi Rama Murthy
Dec 27 '18 at 10:04