Mixing
General definition of Surfaces in $mathbbR^n$
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
What is a definition of a sphere that contains only topological information? And that of a torus? I can only Think of definitions like "points in $mathbbR^n$ that satisfies: $f(x) = 0$" but is there a more General way of defining at Least some Surfaces? Let's take $n=3$ as a reference.
general-topology surfaces spheres
edited Jul 21 at 1:41
Mateus Rocha
446114
asked Jul 21 at 0:29
user199710
212
add a comment |Â
up vote
2
down vote
favorite
What is a definition of a sphere that contains only topological information? And that of a torus? I can only Think of definitions like "points in $mathbbR^n$ that satisfies: $f(x) = 0$" but is there a more General way of defining at Least some Surfaces? Let's take $n=3$ as a reference.
general-topology surfaces spheres
edited Jul 21 at 1:41
Mateus Rocha
446114
asked Jul 21 at 0:29
user199710
212
If you're not talking about specific shapes (which is geometry, and not topology), then you need to specify a sphere as a compact surface of genus $0$ and a torus as a compact surface of genus $1$, etc. But generally shapes have more information than just their underlying topology.
– Ted Shifrin
Jul 21 at 0:36
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 21 at 0:39
Thank you! So what would be like this definition? Can you give me some references?
– user199710
Jul 21 at 9:17
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
What is a definition of a sphere that contains only topological information? And that of a torus? I can only Think of definitions like "points in $mathbbR^n$ that satisfies: $f(x) = 0$" but is there a more General way of defining at Least some Surfaces? Let's take $n=3$ as a reference.
general-topology surfaces spheres
edited Jul 21 at 1:41
Mateus Rocha
446114
asked Jul 21 at 0:29
user199710
212
What is a definition of a sphere that contains only topological information? And that of a torus? I can only Think of definitions like "points in $mathbbR^n$ that satisfies: $f(x) = 0$" but is there a more General way of defining at Least some Surfaces? Let's take $n=3$ as a reference.
general-topology surfaces spheres
edited Jul 21 at 1:41
Mateus Rocha
446114
asked Jul 21 at 0:29
user199710
212
edited Jul 21 at 1:41
Mateus Rocha
446114
edited Jul 21 at 1:41
Mateus Rocha
446114
edited Jul 21 at 1:41
Mateus Rocha
446114
446114
asked Jul 21 at 0:29
user199710
212
asked Jul 21 at 0:29
user199710
212
asked Jul 21 at 0:29
user199710
212
212
If you're not talking about specific shapes (which is geometry, and not topology), then you need to specify a sphere as a compact surface of genus $0$ and a torus as a compact surface of genus $1$, etc. But generally shapes have more information than just their underlying topology.
– Ted Shifrin
Jul 21 at 0:36
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 21 at 0:39
Thank you! So what would be like this definition? Can you give me some references?
– user199710
Jul 21 at 9:17
add a comment |Â
If you're not talking about specific shapes (which is geometry, and not topology), then you need to specify a sphere as a compact surface of genus $0$ and a torus as a compact surface of genus $1$, etc. But generally shapes have more information than just their underlying topology.
– Ted Shifrin
Jul 21 at 0:36
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 21 at 0:39
Thank you! So what would be like this definition? Can you give me some references?
– user199710
Jul 21 at 9:17
If you're not talking about specific shapes (which is geometry, and not topology), then you need to specify a sphere as a compact surface of genus $0$ and a torus as a compact surface of genus $1$, etc. But generally shapes have more information than just their underlying topology.
– Ted Shifrin
Jul 21 at 0:36
If you're not talking about specific shapes (which is geometry, and not topology), then you need to specify a sphere as a compact surface of genus $0$ and a torus as a compact surface of genus $1$, etc. But generally shapes have more information than just their underlying topology.
– Ted Shifrin
Jul 21 at 0:36
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 21 at 0:39
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 21 at 0:39
Thank you! So what would be like this definition? Can you give me some references?
– user199710
Jul 21 at 9:17
Thank you! So what would be like this definition? Can you give me some references?
– user199710
Jul 21 at 9:17
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
1
down vote
The term "surface" can be defined in subtly different ways depending on the context. Two definitions that seem useful to your purposes are:
A topological $2$-manifold (without boundary) in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$.
A topological $2$-manifold with boundary in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$ or to the half-plane $textH^2=(x,y)in mathbb R^2: y geq 0$.
Sometimes these definitions can be modified and restricted $-$ for example, it's quite common to demand that both the homeomorphism and its inverse be smooth functions, in which case $M$ is called a smooth manifold rather than a topological manifold.
Classifying a particular surface $M$ as a sphere, torus, disc, etc. usually involves treating $M$ as a topological subspace of $mathbb R^n$ and describing some of the topological invariants of that space in the hope of classifying $M$ up to homeomorphism or up to homotopy equivalence, which is a weaker condition (homeomorphic spaces are always homotopy equivalent, but not vice versa).
Some of these invariants are probably familiar from general topology $-$ e.g. a distinction is often made between compact and non-compact manifolds. Others are less likely to be familiar. The most important ones are probably the homotopy and homology groups associated with $M$, which together characterize it quite extensively. Unfortunately, the definitions are often quite complex and laboured. However, they are closely associated with simpler invariants, such as the genus and the Euler characteristic. The definition of the Euler characteristic, in particular, is both relatively accessible and of considerable historical importance.
One can also talk about $M$ as a topological space on its own without having it inherit a subspace topology from $mathbb R^n$. This takes more work, but it's possible to topologically characterize surfaces without worrying about what space they're embedded into.
There is quite a lot to say about topological classification of manifolds and entire fields of study dedicated mostly to this topic, so I'm only barely scratching the surface here (excuse the pun). But I hope it is a least a little helpful as a jumping off point.
answered Jul 24 at 15:55
namsos
806
add a comment |Â
up vote
0
down vote
A sphere would be a compact topological manifold of dimension 2 and genus 0. A torus genus 1. These terms you can look up at wikipedia.
answered Jul 23 at 23:45
KALLE THE BAWSMAN
104213
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
The term "surface" can be defined in subtly different ways depending on the context. Two definitions that seem useful to your purposes are:
A topological $2$-manifold (without boundary) in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$.
A topological $2$-manifold with boundary in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$ or to the half-plane $textH^2=(x,y)in mathbb R^2: y geq 0$.
Sometimes these definitions can be modified and restricted $-$ for example, it's quite common to demand that both the homeomorphism and its inverse be smooth functions, in which case $M$ is called a smooth manifold rather than a topological manifold.
Classifying a particular surface $M$ as a sphere, torus, disc, etc. usually involves treating $M$ as a topological subspace of $mathbb R^n$ and describing some of the topological invariants of that space in the hope of classifying $M$ up to homeomorphism or up to homotopy equivalence, which is a weaker condition (homeomorphic spaces are always homotopy equivalent, but not vice versa).
Some of these invariants are probably familiar from general topology $-$ e.g. a distinction is often made between compact and non-compact manifolds. Others are less likely to be familiar. The most important ones are probably the homotopy and homology groups associated with $M$, which together characterize it quite extensively. Unfortunately, the definitions are often quite complex and laboured. However, they are closely associated with simpler invariants, such as the genus and the Euler characteristic. The definition of the Euler characteristic, in particular, is both relatively accessible and of considerable historical importance.
One can also talk about $M$ as a topological space on its own without having it inherit a subspace topology from $mathbb R^n$. This takes more work, but it's possible to topologically characterize surfaces without worrying about what space they're embedded into.
There is quite a lot to say about topological classification of manifolds and entire fields of study dedicated mostly to this topic, so I'm only barely scratching the surface here (excuse the pun). But I hope it is a least a little helpful as a jumping off point.
answered Jul 24 at 15:55
namsos
806
add a comment |Â
up vote
1
down vote
The term "surface" can be defined in subtly different ways depending on the context. Two definitions that seem useful to your purposes are:
A topological $2$-manifold (without boundary) in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$.
A topological $2$-manifold with boundary in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$ or to the half-plane $textH^2=(x,y)in mathbb R^2: y geq 0$.
Sometimes these definitions can be modified and restricted $-$ for example, it's quite common to demand that both the homeomorphism and its inverse be smooth functions, in which case $M$ is called a smooth manifold rather than a topological manifold.
Classifying a particular surface $M$ as a sphere, torus, disc, etc. usually involves treating $M$ as a topological subspace of $mathbb R^n$ and describing some of the topological invariants of that space in the hope of classifying $M$ up to homeomorphism or up to homotopy equivalence, which is a weaker condition (homeomorphic spaces are always homotopy equivalent, but not vice versa).
Some of these invariants are probably familiar from general topology $-$ e.g. a distinction is often made between compact and non-compact manifolds. Others are less likely to be familiar. The most important ones are probably the homotopy and homology groups associated with $M$, which together characterize it quite extensively. Unfortunately, the definitions are often quite complex and laboured. However, they are closely associated with simpler invariants, such as the genus and the Euler characteristic. The definition of the Euler characteristic, in particular, is both relatively accessible and of considerable historical importance.
One can also talk about $M$ as a topological space on its own without having it inherit a subspace topology from $mathbb R^n$. This takes more work, but it's possible to topologically characterize surfaces without worrying about what space they're embedded into.
There is quite a lot to say about topological classification of manifolds and entire fields of study dedicated mostly to this topic, so I'm only barely scratching the surface here (excuse the pun). But I hope it is a least a little helpful as a jumping off point.
answered Jul 24 at 15:55
namsos
806
add a comment |Â
up vote
1
down vote
up vote
1
down vote
The term "surface" can be defined in subtly different ways depending on the context. Two definitions that seem useful to your purposes are:
A topological $2$-manifold (without boundary) in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$.
A topological $2$-manifold with boundary in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$ or to the half-plane $textH^2=(x,y)in mathbb R^2: y geq 0$.
Sometimes these definitions can be modified and restricted $-$ for example, it's quite common to demand that both the homeomorphism and its inverse be smooth functions, in which case $M$ is called a smooth manifold rather than a topological manifold.
Classifying a particular surface $M$ as a sphere, torus, disc, etc. usually involves treating $M$ as a topological subspace of $mathbb R^n$ and describing some of the topological invariants of that space in the hope of classifying $M$ up to homeomorphism or up to homotopy equivalence, which is a weaker condition (homeomorphic spaces are always homotopy equivalent, but not vice versa).
Some of these invariants are probably familiar from general topology $-$ e.g. a distinction is often made between compact and non-compact manifolds. Others are less likely to be familiar. The most important ones are probably the homotopy and homology groups associated with $M$, which together characterize it quite extensively. Unfortunately, the definitions are often quite complex and laboured. However, they are closely associated with simpler invariants, such as the genus and the Euler characteristic. The definition of the Euler characteristic, in particular, is both relatively accessible and of considerable historical importance.
One can also talk about $M$ as a topological space on its own without having it inherit a subspace topology from $mathbb R^n$. This takes more work, but it's possible to topologically characterize surfaces without worrying about what space they're embedded into.
There is quite a lot to say about topological classification of manifolds and entire fields of study dedicated mostly to this topic, so I'm only barely scratching the surface here (excuse the pun). But I hope it is a least a little helpful as a jumping off point.
answered Jul 24 at 15:55
namsos
806
The term "surface" can be defined in subtly different ways depending on the context. Two definitions that seem useful to your purposes are:
A topological $2$-manifold (without boundary) in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$.
A topological $2$-manifold with boundary in $mathbb R^n$ is a set $Msubset mathbb R^n$ such that, for all $p in M$, there exists an open set $U$ such that $pin U$ and $Ucap M$ is homeomorphic to $mathbb R^2$ or to the half-plane $textH^2=(x,y)in mathbb R^2: y geq 0$.
Sometimes these definitions can be modified and restricted $-$ for example, it's quite common to demand that both the homeomorphism and its inverse be smooth functions, in which case $M$ is called a smooth manifold rather than a topological manifold.
Classifying a particular surface $M$ as a sphere, torus, disc, etc. usually involves treating $M$ as a topological subspace of $mathbb R^n$ and describing some of the topological invariants of that space in the hope of classifying $M$ up to homeomorphism or up to homotopy equivalence, which is a weaker condition (homeomorphic spaces are always homotopy equivalent, but not vice versa).
Some of these invariants are probably familiar from general topology $-$ e.g. a distinction is often made between compact and non-compact manifolds. Others are less likely to be familiar. The most important ones are probably the homotopy and homology groups associated with $M$, which together characterize it quite extensively. Unfortunately, the definitions are often quite complex and laboured. However, they are closely associated with simpler invariants, such as the genus and the Euler characteristic. The definition of the Euler characteristic, in particular, is both relatively accessible and of considerable historical importance.
One can also talk about $M$ as a topological space on its own without having it inherit a subspace topology from $mathbb R^n$. This takes more work, but it's possible to topologically characterize surfaces without worrying about what space they're embedded into.
There is quite a lot to say about topological classification of manifolds and entire fields of study dedicated mostly to this topic, so I'm only barely scratching the surface here (excuse the pun). But I hope it is a least a little helpful as a jumping off point.
answered Jul 24 at 15:55
namsos
806
edited Jul 24 at 16:22
answered Jul 24 at 15:55
namsos
806
answered Jul 24 at 15:55
namsos
806
answered Jul 24 at 15:55
namsos
806
806
add a comment |Â
add a comment |Â
up vote
0
down vote
A sphere would be a compact topological manifold of dimension 2 and genus 0. A torus genus 1. These terms you can look up at wikipedia.
answered Jul 23 at 23:45
KALLE THE BAWSMAN
104213
add a comment |Â
up vote
0
down vote
A sphere would be a compact topological manifold of dimension 2 and genus 0. A torus genus 1. These terms you can look up at wikipedia.
answered Jul 23 at 23:45
KALLE THE BAWSMAN
104213
add a comment |Â
up vote
0
down vote
up vote
0
down vote
A sphere would be a compact topological manifold of dimension 2 and genus 0. A torus genus 1. These terms you can look up at wikipedia.
answered Jul 23 at 23:45
KALLE THE BAWSMAN
104213
A sphere would be a compact topological manifold of dimension 2 and genus 0. A torus genus 1. These terms you can look up at wikipedia.
answered Jul 23 at 23:45
KALLE THE BAWSMAN
104213
answered Jul 23 at 23:45
KALLE THE BAWSMAN
104213
answered Jul 23 at 23:45
KALLE THE BAWSMAN
104213
answered Jul 23 at 23:45
KALLE THE BAWSMAN
104213
104213
add a comment |Â
add a comment |Â
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2858130%2fgeneral-definition-of-surfaces-in-mathbbrn%23new-answer', 'question_page');
);
Post as a guest
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
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
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
If you're not talking about specific shapes (which is geometry, and not topology), then you need to specify a sphere as a compact surface of genus $0$ and a torus as a compact surface of genus $1$, etc. But generally shapes have more information than just their underlying topology.
– Ted Shifrin
Jul 21 at 0:36
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 21 at 0:39
Thank you! So what would be like this definition? Can you give me some references?
– user199710
Jul 21 at 9:17