What is {empty set} x something with the product topology?
The product topology is defined as the topology induced by the basis of the product of open sets from each of the original topologies.
From what I understand, then ${emptyset}x]a,b[$ is an open set of $IR^2$ with the product topology, right? But what is this geometrically? I can't see what this is.
general-topology product-space
edited 2 days ago
Lehs
6,95231662
asked 2 days ago
J. DionisioJ. Dionisio
9511
add a comment |
The product topology is defined as the topology induced by the basis of the product of open sets from each of the original topologies.
From what I understand, then ${emptyset}x]a,b[$ is an open set of $IR^2$ with the product topology, right? But what is this geometrically? I can't see what this is.
general-topology product-space
edited 2 days ago
Lehs
6,95231662
asked 2 days ago
J. DionisioJ. Dionisio
9511
3
Empty cross anything is empty.
– Randall
2 days ago
The set ${ emptyset }$ has at most one topology. For any space $A$ you can show ${emptyset } times A$ is homeomorphic to $A$
– leibnewtz
2 days ago
Being "homeomorphic to" is intuitively the same as "looking like," so it just looks like $mathbb{R}$. I should say that you're wrong in assuming ${emptyset }$ is open. And thanks!
– leibnewtz
2 days ago
add a comment |
The product topology is defined as the topology induced by the basis of the product of open sets from each of the original topologies.
From what I understand, then ${emptyset}x]a,b[$ is an open set of $IR^2$ with the product topology, right? But what is this geometrically? I can't see what this is.
general-topology product-space
edited 2 days ago
Lehs
6,95231662
asked 2 days ago
J. DionisioJ. Dionisio
9511
The product topology is defined as the topology induced by the basis of the product of open sets from each of the original topologies.
From what I understand, then ${emptyset}x]a,b[$ is an open set of $IR^2$ with the product topology, right? But what is this geometrically? I can't see what this is.
general-topology product-space
general-topology product-space
edited 2 days ago
Lehs
6,95231662
asked 2 days ago
J. DionisioJ. Dionisio
9511
edited 2 days ago
Lehs
6,95231662
asked 2 days ago
J. DionisioJ. Dionisio
9511
edited 2 days ago
Lehs
6,95231662
edited 2 days ago
Lehs
6,95231662
edited 2 days ago
Lehs
6,95231662
6,95231662
asked 2 days ago
J. DionisioJ. Dionisio
9511
asked 2 days ago
J. DionisioJ. Dionisio
9511
asked 2 days ago
J. DionisioJ. Dionisio
9511
9511
3
Empty cross anything is empty.
– Randall
2 days ago
The set ${ emptyset }$ has at most one topology. For any space $A$ you can show ${emptyset } times A$ is homeomorphic to $A$
– leibnewtz
2 days ago
Being "homeomorphic to" is intuitively the same as "looking like," so it just looks like $mathbb{R}$. I should say that you're wrong in assuming ${emptyset }$ is open. And thanks!
– leibnewtz
2 days ago
add a comment |
3
Empty cross anything is empty.
– Randall
2 days ago
The set ${ emptyset }$ has at most one topology. For any space $A$ you can show ${emptyset } times A$ is homeomorphic to $A$
– leibnewtz
2 days ago
Being "homeomorphic to" is intuitively the same as "looking like," so it just looks like $mathbb{R}$. I should say that you're wrong in assuming ${emptyset }$ is open. And thanks!
– leibnewtz
2 days ago
3
3
Empty cross anything is empty.
– Randall
2 days ago
Empty cross anything is empty.
– Randall
2 days ago
The set ${ emptyset }$ has at most one topology. For any space $A$ you can show ${emptyset } times A$ is homeomorphic to $A$
– leibnewtz
2 days ago
The set ${ emptyset }$ has at most one topology. For any space $A$ you can show ${emptyset } times A$ is homeomorphic to $A$
– leibnewtz
2 days ago
Being "homeomorphic to" is intuitively the same as "looking like," so it just looks like $mathbb{R}$. I should say that you're wrong in assuming ${emptyset }$ is open. And thanks!
– leibnewtz
2 days ago
Being "homeomorphic to" is intuitively the same as "looking like," so it just looks like $mathbb{R}$. I should say that you're wrong in assuming ${emptyset }$ is open. And thanks!
– leibnewtz
2 days ago
add a comment |
2 Answers
2
active
oldest
votes
By definition, the elements of $Atimes B$ (for sets $A$ and $B$) are the ordered pairs $langle a,brangle$ such that $ain A$ and $bin B.$ Since $ainemptyset$ is impossible, then $emptysettimes B$ has no elements, meaning $emptysettimes B=emptyset,$ regardless of the set $B.$
The open sets of $Bbb R^2$ in the product topology can be obtained from the basis ${Atimes B:A,Btext{ are open subsets of }Bbb R}.$ This will include elements of the form $emptysettimes B=emptyset,$ as well as $Atimesemptyset=emptyset.$ However, it will not have any elements of the form ${emptyset}times B,$ since $emptyset$ isn't an element of $Bbb R.$
answered 2 days ago
Cameron BuieCameron Buie
85.1k771155
The question is asking about ${emptyset } times A$, not $emptyset times A$
– leibnewtz
2 days ago
Maybe I didn't understand your answer, but just to be sure, ∅ x O, where O is an open set of the topology on B is an open set of the product topology on A x B, right? Different from U x ∅, where U is an open set of the topology on B
– J. Dionisio
2 days ago
Oh, dear! I misunderstood. I'll edit accordingly.
– Cameron Buie
2 days ago
Ohhh okay, now I understand, thank you! I was mistaking ∅ with {∅}. Thanks a lot for the help!
– J. Dionisio
2 days ago
add a comment |
Instead of ${emptyset}$, I'll consider ${*}$ just to avoid confusion. It really doesn't matter, you can use anything instead of $*$. What matters is that ${*}$ is a singleton as a set.
Let $X$ be a topological space. Then, basis for topology of ${*}times X$ is given by ${ {*}times Umid Usubseteq X text{open}}$.
Define $fcolon Xto {*}times X$ by $f(x) = (*,x)$ and $gcolon {*}times Xto X$ by $g(*,x) = x$. Obviously, these functions are inverses of each other. They are also continuous, since for any $Usubseteq X$ open we have
$$f^{-1}({*}times U) = U, g^{-1}(U) = {*}times U.$$
Thus, $X$ and ${*}times X$ are homeomorphic.
If $X = mathbb R$, instead of $*$ choose some real number $a$. Then ${a}timesmathbb R$ is naturally a line embedded in $mathbb R^2$. Here are examples for some different $a$'s:
answered 2 days ago
EnnarEnnar
14.4k32343
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%2f3063120%2fwhat-is-empty-set-x-something-with-the-product-topology%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
By definition, the elements of $Atimes B$ (for sets $A$ and $B$) are the ordered pairs $langle a,brangle$ such that $ain A$ and $bin B.$ Since $ainemptyset$ is impossible, then $emptysettimes B$ has no elements, meaning $emptysettimes B=emptyset,$ regardless of the set $B.$
The open sets of $Bbb R^2$ in the product topology can be obtained from the basis ${Atimes B:A,Btext{ are open subsets of }Bbb R}.$ This will include elements of the form $emptysettimes B=emptyset,$ as well as $Atimesemptyset=emptyset.$ However, it will not have any elements of the form ${emptyset}times B,$ since $emptyset$ isn't an element of $Bbb R.$
answered 2 days ago
Cameron BuieCameron Buie
85.1k771155
The question is asking about ${emptyset } times A$, not $emptyset times A$
– leibnewtz
2 days ago
Maybe I didn't understand your answer, but just to be sure, ∅ x O, where O is an open set of the topology on B is an open set of the product topology on A x B, right? Different from U x ∅, where U is an open set of the topology on B
– J. Dionisio
2 days ago
Oh, dear! I misunderstood. I'll edit accordingly.
– Cameron Buie
2 days ago
Ohhh okay, now I understand, thank you! I was mistaking ∅ with {∅}. Thanks a lot for the help!
– J. Dionisio
2 days ago
add a comment |
By definition, the elements of $Atimes B$ (for sets $A$ and $B$) are the ordered pairs $langle a,brangle$ such that $ain A$ and $bin B.$ Since $ainemptyset$ is impossible, then $emptysettimes B$ has no elements, meaning $emptysettimes B=emptyset,$ regardless of the set $B.$
The open sets of $Bbb R^2$ in the product topology can be obtained from the basis ${Atimes B:A,Btext{ are open subsets of }Bbb R}.$ This will include elements of the form $emptysettimes B=emptyset,$ as well as $Atimesemptyset=emptyset.$ However, it will not have any elements of the form ${emptyset}times B,$ since $emptyset$ isn't an element of $Bbb R.$
answered 2 days ago
Cameron BuieCameron Buie
85.1k771155
The question is asking about ${emptyset } times A$, not $emptyset times A$
– leibnewtz
2 days ago
Maybe I didn't understand your answer, but just to be sure, ∅ x O, where O is an open set of the topology on B is an open set of the product topology on A x B, right? Different from U x ∅, where U is an open set of the topology on B
– J. Dionisio
2 days ago
Oh, dear! I misunderstood. I'll edit accordingly.
– Cameron Buie
2 days ago
Ohhh okay, now I understand, thank you! I was mistaking ∅ with {∅}. Thanks a lot for the help!
– J. Dionisio
2 days ago
add a comment |
By definition, the elements of $Atimes B$ (for sets $A$ and $B$) are the ordered pairs $langle a,brangle$ such that $ain A$ and $bin B.$ Since $ainemptyset$ is impossible, then $emptysettimes B$ has no elements, meaning $emptysettimes B=emptyset,$ regardless of the set $B.$
The open sets of $Bbb R^2$ in the product topology can be obtained from the basis ${Atimes B:A,Btext{ are open subsets of }Bbb R}.$ This will include elements of the form $emptysettimes B=emptyset,$ as well as $Atimesemptyset=emptyset.$ However, it will not have any elements of the form ${emptyset}times B,$ since $emptyset$ isn't an element of $Bbb R.$
answered 2 days ago
Cameron BuieCameron Buie
85.1k771155
By definition, the elements of $Atimes B$ (for sets $A$ and $B$) are the ordered pairs $langle a,brangle$ such that $ain A$ and $bin B.$ Since $ainemptyset$ is impossible, then $emptysettimes B$ has no elements, meaning $emptysettimes B=emptyset,$ regardless of the set $B.$
The open sets of $Bbb R^2$ in the product topology can be obtained from the basis ${Atimes B:A,Btext{ are open subsets of }Bbb R}.$ This will include elements of the form $emptysettimes B=emptyset,$ as well as $Atimesemptyset=emptyset.$ However, it will not have any elements of the form ${emptyset}times B,$ since $emptyset$ isn't an element of $Bbb R.$
answered 2 days ago
Cameron BuieCameron Buie
85.1k771155
edited 2 days ago
answered 2 days ago
Cameron BuieCameron Buie
85.1k771155
answered 2 days ago
Cameron BuieCameron Buie
85.1k771155
answered 2 days ago
Cameron BuieCameron Buie
85.1k771155
85.1k771155
The question is asking about ${emptyset } times A$, not $emptyset times A$
– leibnewtz
2 days ago
Maybe I didn't understand your answer, but just to be sure, ∅ x O, where O is an open set of the topology on B is an open set of the product topology on A x B, right? Different from U x ∅, where U is an open set of the topology on B
– J. Dionisio
2 days ago
Oh, dear! I misunderstood. I'll edit accordingly.
– Cameron Buie
2 days ago
Ohhh okay, now I understand, thank you! I was mistaking ∅ with {∅}. Thanks a lot for the help!
– J. Dionisio
2 days ago
add a comment |
The question is asking about ${emptyset } times A$, not $emptyset times A$
– leibnewtz
2 days ago
Maybe I didn't understand your answer, but just to be sure, ∅ x O, where O is an open set of the topology on B is an open set of the product topology on A x B, right? Different from U x ∅, where U is an open set of the topology on B
– J. Dionisio
2 days ago
Oh, dear! I misunderstood. I'll edit accordingly.
– Cameron Buie
2 days ago
Ohhh okay, now I understand, thank you! I was mistaking ∅ with {∅}. Thanks a lot for the help!
– J. Dionisio
2 days ago
The question is asking about ${emptyset } times A$, not $emptyset times A$
– leibnewtz
2 days ago
The question is asking about ${emptyset } times A$, not $emptyset times A$
– leibnewtz
2 days ago
Maybe I didn't understand your answer, but just to be sure, ∅ x O, where O is an open set of the topology on B is an open set of the product topology on A x B, right? Different from U x ∅, where U is an open set of the topology on B
– J. Dionisio
2 days ago
Maybe I didn't understand your answer, but just to be sure, ∅ x O, where O is an open set of the topology on B is an open set of the product topology on A x B, right? Different from U x ∅, where U is an open set of the topology on B
– J. Dionisio
2 days ago
Oh, dear! I misunderstood. I'll edit accordingly.
– Cameron Buie
2 days ago
Oh, dear! I misunderstood. I'll edit accordingly.
– Cameron Buie
2 days ago
Ohhh okay, now I understand, thank you! I was mistaking ∅ with {∅}. Thanks a lot for the help!
– J. Dionisio
2 days ago
Ohhh okay, now I understand, thank you! I was mistaking ∅ with {∅}. Thanks a lot for the help!
– J. Dionisio
2 days ago
add a comment |
Instead of ${emptyset}$, I'll consider ${*}$ just to avoid confusion. It really doesn't matter, you can use anything instead of $*$. What matters is that ${*}$ is a singleton as a set.
Let $X$ be a topological space. Then, basis for topology of ${*}times X$ is given by ${ {*}times Umid Usubseteq X text{open}}$.
Define $fcolon Xto {*}times X$ by $f(x) = (*,x)$ and $gcolon {*}times Xto X$ by $g(*,x) = x$. Obviously, these functions are inverses of each other. They are also continuous, since for any $Usubseteq X$ open we have
$$f^{-1}({*}times U) = U, g^{-1}(U) = {*}times U.$$
Thus, $X$ and ${*}times X$ are homeomorphic.
If $X = mathbb R$, instead of $*$ choose some real number $a$. Then ${a}timesmathbb R$ is naturally a line embedded in $mathbb R^2$. Here are examples for some different $a$'s:
answered 2 days ago
EnnarEnnar
14.4k32343
add a comment |
Instead of ${emptyset}$, I'll consider ${*}$ just to avoid confusion. It really doesn't matter, you can use anything instead of $*$. What matters is that ${*}$ is a singleton as a set.
Let $X$ be a topological space. Then, basis for topology of ${*}times X$ is given by ${ {*}times Umid Usubseteq X text{open}}$.
Define $fcolon Xto {*}times X$ by $f(x) = (*,x)$ and $gcolon {*}times Xto X$ by $g(*,x) = x$. Obviously, these functions are inverses of each other. They are also continuous, since for any $Usubseteq X$ open we have
$$f^{-1}({*}times U) = U, g^{-1}(U) = {*}times U.$$
Thus, $X$ and ${*}times X$ are homeomorphic.
If $X = mathbb R$, instead of $*$ choose some real number $a$. Then ${a}timesmathbb R$ is naturally a line embedded in $mathbb R^2$. Here are examples for some different $a$'s:
answered 2 days ago
EnnarEnnar
14.4k32343
add a comment |
Instead of ${emptyset}$, I'll consider ${*}$ just to avoid confusion. It really doesn't matter, you can use anything instead of $*$. What matters is that ${*}$ is a singleton as a set.
Let $X$ be a topological space. Then, basis for topology of ${*}times X$ is given by ${ {*}times Umid Usubseteq X text{open}}$.
Define $fcolon Xto {*}times X$ by $f(x) = (*,x)$ and $gcolon {*}times Xto X$ by $g(*,x) = x$. Obviously, these functions are inverses of each other. They are also continuous, since for any $Usubseteq X$ open we have
$$f^{-1}({*}times U) = U, g^{-1}(U) = {*}times U.$$
Thus, $X$ and ${*}times X$ are homeomorphic.
If $X = mathbb R$, instead of $*$ choose some real number $a$. Then ${a}timesmathbb R$ is naturally a line embedded in $mathbb R^2$. Here are examples for some different $a$'s:
answered 2 days ago
EnnarEnnar
14.4k32343
Instead of ${emptyset}$, I'll consider ${*}$ just to avoid confusion. It really doesn't matter, you can use anything instead of $*$. What matters is that ${*}$ is a singleton as a set.
Let $X$ be a topological space. Then, basis for topology of ${*}times X$ is given by ${ {*}times Umid Usubseteq X text{open}}$.
Define $fcolon Xto {*}times X$ by $f(x) = (*,x)$ and $gcolon {*}times Xto X$ by $g(*,x) = x$. Obviously, these functions are inverses of each other. They are also continuous, since for any $Usubseteq X$ open we have
$$f^{-1}({*}times U) = U, g^{-1}(U) = {*}times U.$$
Thus, $X$ and ${*}times X$ are homeomorphic.
If $X = mathbb R$, instead of $*$ choose some real number $a$. Then ${a}timesmathbb R$ is naturally a line embedded in $mathbb R^2$. Here are examples for some different $a$'s:
answered 2 days ago
EnnarEnnar
14.4k32343
edited 2 days ago
answered 2 days ago
EnnarEnnar
14.4k32343
answered 2 days ago
EnnarEnnar
14.4k32343
answered 2 days ago
EnnarEnnar
14.4k32343
14.4k32343
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3063120%2fwhat-is-empty-set-x-something-with-the-product-topology%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
3
Empty cross anything is empty.
– Randall
2 days ago
The set ${ emptyset }$ has at most one topology. For any space $A$ you can show ${emptyset } times A$ is homeomorphic to $A$
– leibnewtz
2 days ago
Being "homeomorphic to" is intuitively the same as "looking like," so it just looks like $mathbb{R}$. I should say that you're wrong in assuming ${emptyset }$ is open. And thanks!
– leibnewtz
2 days ago