Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lemma 13.2 from Munkres

  1. Aug 1, 2011 #1
    Lemma 13.2: Let X be a topological space. Suppose that C is a collection of open sets of X such that for each open set U of X and each x in U, there is an element c of C such that [itex]x\in c\subset U[/itex]. Then C is a basis for the topology of X.

    Proof: The first paragraph is trivial, it just shows that the conditions of basis are satisfied.

    The second paragraph attempts to show that [itex]\tau'[/itex], the topology generated by C, is the same as the topology [itex]\tau[/itex] on X.

    Can someone elaborate on the second paragraph please!?!
     
  2. jcsd
  3. Aug 1, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    What don't you understand about the second paragraph??

    You just let [itex]\mathcal{T}^\prime[/itex] be the topology generated by the basis [itex]\mathcal{C}[/itex]. So by definition, a set G belongs to [itex]\mathcal{T}^\prime[/itex] if for every x in G, there is a set C in [itex]\mathcal{C}[/itex] such that

    [tex]x\in C\subseteq G[/tex]

    So, to prove that [itex]\mathcal{T}\subseteq\mathcal{T}^\prime[/itex]. Take G in [itex]\mathcal{T}[/itex]. By hypothesis, there exists for every for every x in G, a set C in [itex]\mathcal{C}[/itex] such that

    [tex]x\in C\subseteq G[/tex]

    So, by definition almost, we have that [itex]\mathcal{T}\subseteq \mathcal{T}^\prime[/itex].

    The other inclusion is less obvious. So to prove that [itex]\mathcal{T}^\prime\subseteq\mathcal{T}[/itex] we take an element [itex]G\in \mathcal{T}^\prime[/itex]. By hypothesis, there is for every x in G, a set C in [itex]\mathcal{C}[/itex] such that

    [tex]x\in C\subseteq G[/tex]

    This implies that [itex]G=\bigcup{C}[/itex] is the union of all these C's. Since all the C's are in [itex]\mathcal{T}[/itex], it implies that G is also in [itex]\mathcal{T}[/itex] as union of open sets...

    IS that more clear?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Lemma 13.2 from Munkres
  1. Munkres text question. (Replies: 1)

Loading...