Mixing
Proving that $displaystyle{{(x_n, y_n) }}_{n in mathbb{N}}$ converges to $(x,y)$ on the product topology iff...
$begingroup$
Let $(Omega_1, tau_1)$ and $(Omega_2, tau_2)$ be two topological spaces. Prove that $displaystyle{{(x_n, y_n) }}_{n in mathbb{N}}$ converges to $(x,y)$ on the product topology if and only if $x_n to x$ in $Omega_1$ and $y_ n to y$ in $Omega_2$.
My proof:
$implies$: Let $U ni x$ and $V ni y$ be open sets of $Omega_1$ and $Omega_2$, respectively. Then $(x, y) in U times V$, so by hypothesis there exists $N in mathbb{N}$ such that $displaystyle{{(x_n, y_n) }}_{n in mathbb{N}} in U times V forall n geq N$. Then $x_n in U$ and $y_n in V$ for all but a finite number of terms and we're done.
$impliedby$: Let $A ni (x, y)$ be an open set of $Omega_1 times Omega_2$. Clearly there exist $B in beta$ and $C in mathfrak{C}$ such that $(x,y) in B times C subset A$, where $beta$ and $mathfrak{C}$ are basis for $Omega_1$ and $Omega_2$, respectively. By hypothesis, for any open sets $Omega_1 supset U ni x$ and $Omega_2 supset V ni y$, there exist $N_1$ and $N_2$ in $mathbb{N}$ such that $x_n in U forall n geq N_1$ and $y_n in V forall n geq N_2$. In particular, this is true for $B$ and $C$. Then: $$displaystyle{{(x_n, y_n) }}_{n in mathbb{N}} in B times C subset A forall n geq max(N_1, N_2).$$
and we're done.
Is all of this alright? It just looks too easy and I wanna make sure I didn't make any mistakes...
sequences-and-series general-topology
asked Jan 16 at 23:26
Matheus AndradeMatheus Andrade
1,374418
$endgroup$
add a comment |
$begingroup$
Let $(Omega_1, tau_1)$ and $(Omega_2, tau_2)$ be two topological spaces. Prove that $displaystyle{{(x_n, y_n) }}_{n in mathbb{N}}$ converges to $(x,y)$ on the product topology if and only if $x_n to x$ in $Omega_1$ and $y_ n to y$ in $Omega_2$.
My proof:
$implies$: Let $U ni x$ and $V ni y$ be open sets of $Omega_1$ and $Omega_2$, respectively. Then $(x, y) in U times V$, so by hypothesis there exists $N in mathbb{N}$ such that $displaystyle{{(x_n, y_n) }}_{n in mathbb{N}} in U times V forall n geq N$. Then $x_n in U$ and $y_n in V$ for all but a finite number of terms and we're done.
$impliedby$: Let $A ni (x, y)$ be an open set of $Omega_1 times Omega_2$. Clearly there exist $B in beta$ and $C in mathfrak{C}$ such that $(x,y) in B times C subset A$, where $beta$ and $mathfrak{C}$ are basis for $Omega_1$ and $Omega_2$, respectively. By hypothesis, for any open sets $Omega_1 supset U ni x$ and $Omega_2 supset V ni y$, there exist $N_1$ and $N_2$ in $mathbb{N}$ such that $x_n in U forall n geq N_1$ and $y_n in V forall n geq N_2$. In particular, this is true for $B$ and $C$. Then: $$displaystyle{{(x_n, y_n) }}_{n in mathbb{N}} in B times C subset A forall n geq max(N_1, N_2).$$
and we're done.
Is all of this alright? It just looks too easy and I wanna make sure I didn't make any mistakes...
sequences-and-series general-topology
asked Jan 16 at 23:26
Matheus AndradeMatheus Andrade
1,374418
$endgroup$
1
$begingroup$
Your proof is fine.
$endgroup$
– Kavi Rama Murthy
Jan 16 at 23:28
1
$begingroup$
@KaviRamaMurthy Thanks!
$endgroup$
– Matheus Andrade
Jan 16 at 23:30
$begingroup$
First part. The projections are continuous and continuous functions preserve limits.
$endgroup$
– William Elliot
Jan 17 at 7:14
add a comment |
$begingroup$
Let $(Omega_1, tau_1)$ and $(Omega_2, tau_2)$ be two topological spaces. Prove that $displaystyle{{(x_n, y_n) }}_{n in mathbb{N}}$ converges to $(x,y)$ on the product topology if and only if $x_n to x$ in $Omega_1$ and $y_ n to y$ in $Omega_2$.
My proof:
$implies$: Let $U ni x$ and $V ni y$ be open sets of $Omega_1$ and $Omega_2$, respectively. Then $(x, y) in U times V$, so by hypothesis there exists $N in mathbb{N}$ such that $displaystyle{{(x_n, y_n) }}_{n in mathbb{N}} in U times V forall n geq N$. Then $x_n in U$ and $y_n in V$ for all but a finite number of terms and we're done.
$impliedby$: Let $A ni (x, y)$ be an open set of $Omega_1 times Omega_2$. Clearly there exist $B in beta$ and $C in mathfrak{C}$ such that $(x,y) in B times C subset A$, where $beta$ and $mathfrak{C}$ are basis for $Omega_1$ and $Omega_2$, respectively. By hypothesis, for any open sets $Omega_1 supset U ni x$ and $Omega_2 supset V ni y$, there exist $N_1$ and $N_2$ in $mathbb{N}$ such that $x_n in U forall n geq N_1$ and $y_n in V forall n geq N_2$. In particular, this is true for $B$ and $C$. Then: $$displaystyle{{(x_n, y_n) }}_{n in mathbb{N}} in B times C subset A forall n geq max(N_1, N_2).$$
and we're done.
Is all of this alright? It just looks too easy and I wanna make sure I didn't make any mistakes...
sequences-and-series general-topology
asked Jan 16 at 23:26
Matheus AndradeMatheus Andrade
1,374418
$endgroup$
Let $(Omega_1, tau_1)$ and $(Omega_2, tau_2)$ be two topological spaces. Prove that $displaystyle{{(x_n, y_n) }}_{n in mathbb{N}}$ converges to $(x,y)$ on the product topology if and only if $x_n to x$ in $Omega_1$ and $y_ n to y$ in $Omega_2$.
My proof:
$implies$: Let $U ni x$ and $V ni y$ be open sets of $Omega_1$ and $Omega_2$, respectively. Then $(x, y) in U times V$, so by hypothesis there exists $N in mathbb{N}$ such that $displaystyle{{(x_n, y_n) }}_{n in mathbb{N}} in U times V forall n geq N$. Then $x_n in U$ and $y_n in V$ for all but a finite number of terms and we're done.
$impliedby$: Let $A ni (x, y)$ be an open set of $Omega_1 times Omega_2$. Clearly there exist $B in beta$ and $C in mathfrak{C}$ such that $(x,y) in B times C subset A$, where $beta$ and $mathfrak{C}$ are basis for $Omega_1$ and $Omega_2$, respectively. By hypothesis, for any open sets $Omega_1 supset U ni x$ and $Omega_2 supset V ni y$, there exist $N_1$ and $N_2$ in $mathbb{N}$ such that $x_n in U forall n geq N_1$ and $y_n in V forall n geq N_2$. In particular, this is true for $B$ and $C$. Then: $$displaystyle{{(x_n, y_n) }}_{n in mathbb{N}} in B times C subset A forall n geq max(N_1, N_2).$$
and we're done.
Is all of this alright? It just looks too easy and I wanna make sure I didn't make any mistakes...
sequences-and-series general-topology
sequences-and-series general-topology
asked Jan 16 at 23:26
Matheus AndradeMatheus Andrade
1,374418
asked Jan 16 at 23:26
Matheus AndradeMatheus Andrade
1,374418
asked Jan 16 at 23:26
Matheus AndradeMatheus Andrade
1,374418
asked Jan 16 at 23:26
Matheus AndradeMatheus Andrade
1,374418
asked Jan 16 at 23:26
Matheus AndradeMatheus Andrade
1,374418
1,374418
1
$begingroup$
Your proof is fine.
$endgroup$
– Kavi Rama Murthy
Jan 16 at 23:28
1
$begingroup$
@KaviRamaMurthy Thanks!
$endgroup$
– Matheus Andrade
Jan 16 at 23:30
$begingroup$
First part. The projections are continuous and continuous functions preserve limits.
$endgroup$
– William Elliot
Jan 17 at 7:14
add a comment |
1
$begingroup$
Your proof is fine.
$endgroup$
– Kavi Rama Murthy
Jan 16 at 23:28
1
$begingroup$
@KaviRamaMurthy Thanks!
$endgroup$
– Matheus Andrade
Jan 16 at 23:30
$begingroup$
First part. The projections are continuous and continuous functions preserve limits.
$endgroup$
– William Elliot
Jan 17 at 7:14
1
1
$begingroup$
Your proof is fine.
$endgroup$
– Kavi Rama Murthy
Jan 16 at 23:28
$begingroup$
Your proof is fine.
$endgroup$
– Kavi Rama Murthy
Jan 16 at 23:28
1
1
$begingroup$
@KaviRamaMurthy Thanks!
$endgroup$
– Matheus Andrade
Jan 16 at 23:30
$begingroup$
@KaviRamaMurthy Thanks!
$endgroup$
– Matheus Andrade
Jan 16 at 23:30
$begingroup$
First part. The projections are continuous and continuous functions preserve limits.
$endgroup$
– William Elliot
Jan 17 at 7:14
$begingroup$
First part. The projections are continuous and continuous functions preserve limits.
$endgroup$
– William Elliot
Jan 17 at 7:14
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The proof I gave is indeed correct.
answered Jan 18 at 14:42
Matheus AndradeMatheus Andrade
1,374418
$endgroup$
add a comment |
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3076422%2fproving-that-displaystyle-x-n-y-n-n-in-mathbbn-converges-to%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
$begingroup$
The proof I gave is indeed correct.
answered Jan 18 at 14:42
Matheus AndradeMatheus Andrade
1,374418
$endgroup$
add a comment |
$begingroup$
The proof I gave is indeed correct.
answered Jan 18 at 14:42
Matheus AndradeMatheus Andrade
1,374418
$endgroup$
add a comment |
$begingroup$
The proof I gave is indeed correct.
answered Jan 18 at 14:42
Matheus AndradeMatheus Andrade
1,374418
$endgroup$
The proof I gave is indeed correct.
answered Jan 18 at 14:42
Matheus AndradeMatheus Andrade
1,374418
answered Jan 18 at 14:42
Matheus AndradeMatheus Andrade
1,374418
answered Jan 18 at 14:42
Matheus AndradeMatheus Andrade
1,374418
answered Jan 18 at 14:42
Matheus AndradeMatheus Andrade
1,374418
1,374418
add a comment |
add a comment |
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3076422%2fproving-that-displaystyle-x-n-y-n-n-in-mathbbn-converges-to%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
1
$begingroup$
Your proof is fine.
$endgroup$
– Kavi Rama Murthy
Jan 16 at 23:28
1
$begingroup$
@KaviRamaMurthy Thanks!
$endgroup$
– Matheus Andrade
Jan 16 at 23:30
$begingroup$
First part. The projections are continuous and continuous functions preserve limits.
$endgroup$
– William Elliot
Jan 17 at 7:14