Results 1 to 6 of 6

Math Help - What is "a point at infinity"?

  1. #1
    ynj
    ynj is offline
    Senior Member
    Joined
    Jul 2009
    Posts
    254

    What is "a point at infinity"?

    What is "a point at infinity" in the complex field?
    What is the neighbourhood of a point at infinity?
    What does it mean if a connected set contains "a point at infinity"?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Feb 2010
    From
    Lisbon
    Posts
    51

    Riemann Sphere

    If I understand correctly, you're asking what \mathbb C \cup \{ \infty\} is (called the Riemann sphere). As a set it's simply the set of complex numbers \mathbb C together with an additional point you call "infinity" and denote by \infty. As a topological space, it's the one-point compactification of the complex plane: a set is open in \mathbb C \cup \{ \infty\} iff it's open in \mathbb C or is the complement of a compact set (in \mathbb C).

    This definition should be enough for you to answer the two other questions.

    Intuitively, imagine you grab the real line and wrap it around a circle. It's "natural" to say the line won't cover the whole circle: there will be one point missing, so you call this point infinity. Same goes for the complex plane folded around a sphere: one point of the sphere is missing, so you just cover the hole with an extra "point at infinity". That's why it's called the Riemann sphere.

    Hope it helped
    Follow Math Help Forum on Facebook and Google+

  3. #3
    ynj
    ynj is offline
    Senior Member
    Joined
    Jul 2009
    Posts
    254
    Quote Originally Posted by Nyrox View Post
    If I understand correctly, you're asking what \mathbb C \cup \{ \infty\} is (called the Riemann sphere). As a set it's simply the set of complex numbers \mathbb C together with an additional point you call "infinity" and denote by \infty. As a topological space, it's the one-point compactification of the complex plane: a set is open in \mathbb C \cup \{ \infty\} iff it's open in \mathbb C or is the complement of a compact set (in \mathbb C).

    This definition should be enough for you to answer the two other questions.

    Intuitively, imagine you grab the real line and wrap it around a circle. It's "natural" to say the line won't cover the whole circle: there will be one point missing, so you call this point infinity. Same goes for the complex plane folded around a sphere: one point of the sphere is missing, so you just cover the hole with an extra "point at infinity". That's why it's called the Riemann sphere.

    Hope it helped
    Hmm...But if a set is compact, then it implies that it is closed and bounded, hence its complement is open.....Any thing difference from the original definition of open set in \mathbb{c}?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Feb 2010
    From
    Lisbon
    Posts
    51
    By "the complement of a compact set (in \mathbb C)" I mean the complement (in \mathbb C \cup \{ \infty\}) of a compact (in \mathbb C). For instance:

    \{z\in\mathbb C:|z|>2\}\cup\{\infty\}

    is an open set in \mathbb C\cup\{\infty\} since its complement is the closed ball of radius 2 centered at the origin, which is compact.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    ynj
    ynj is offline
    Senior Member
    Joined
    Jul 2009
    Posts
    254
    Quote Originally Posted by Nyrox View Post
    By "the complement of a compact set (in \mathbb C)" I mean the complement (in \mathbb C \cup \{ \infty\}) of a compact (in \mathbb C). For instance:

    \{z\in\mathbb C:|z|>2\}\cup\{\infty\}

    is an open set in \mathbb C\cup\{\infty\} since its complement is the closed ball of radius 2 centered at the origin, which is compact.
    Then is it right if I say the connectedness is equivalent to "cannot be the union of two nonempty disjoint relatively open set in the extended plane"?
    Last edited by ynj; February 6th 2010 at 08:05 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Feb 2010
    From
    Lisbon
    Posts
    51
    Not exactly relatively open (although you can think of it like so in practice). It's connected if it cannot be written as the union of two nonempty disjoint open sets in this new topology we defined. For instance, my previous example:

    \{z\in\mathbb C:|z|>2\}\cup\{\infty\}

    is connected in \mathbb C\cup \{\infty\}. But not in \mathbb C: it's not even a subset of \mathbb C! In practice, though, one can "ignore" the point at infinity (for connectedness related issues) and just think of it as a set in \mathbb C, in which case your description is, I believe, valid.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: October 17th 2011, 03:50 PM
  2. Infinity and "Lost in Hyperspace"
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: July 18th 2011, 05:59 AM
  3. Replies: 2
    Last Post: April 24th 2011, 08:01 AM
  4. Replies: 3
    Last Post: November 18th 2010, 04:21 AM
  5. Replies: 1
    Last Post: October 25th 2010, 05:45 AM

Search Tags


/mathhelpforum @mathhelpforum