Continuity of two variables in toplogical space .












1












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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
















1












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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














1












1








1





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 27 '18 at 8:50









MotylaNogaTomkaMazura

6,597917




6,597917










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


















  • $begingroup$
    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










1 Answer
1






active

oldest

votes


















0












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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











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1 Answer
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1 Answer
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active

oldest

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active

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0












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
















0












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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














0












0








0





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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


















  • $begingroup$
    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


















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