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

6,572917




6,572917










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











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053701%2fcontinuity-of-two-variables-in-toplogical-space%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






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
















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

60.5k42161




60.5k42161












  • $begingroup$
    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


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053701%2fcontinuity-of-two-variables-in-toplogical-space%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Bressuire

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more

Cabo Verde

Image
  Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
Read more

Gyllenstierna

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more