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Hausdorff and Non-Hausdorff Spaces


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Hausdorff and Non-Hausdorff Spaces

Hausdorff and Non-Hausdorff Spaces

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  • 1. Hausdorff and Non-Hausdorff Spaces Mayra Ibarra MATH101
  • 2. What is a Topological Space?
    • Recall:
      • A topological space is an ordered pair (X, Ƭ), where X is a set and T is a collection of subsets of X such that
        • Ø Є Ƭ and X Є Ƭ
        • U ∩V Є Ƭ whenever U, V Є Ƭ ;
        • ∪ { U a : a Є I } Є Ƭ whenever {U a : a Є I } ⊂ Ƭ. 1
  • 3. Separation Axioms 2
    • T 0 : If a, b ϵ X, there exists an open set O ϵ Ƭ such that either a ϵ O and b O, or b ϵ O and a O.
    • T 1 : If a, b ϵ X, there exist open sets O a , O b ϵ Ƭ containing a and b respectively, such that b O a and a O b .
  • 4. T 2 : If a, b ϵ X, there exist disjoint open sets O a and O b containing a and b respectively.
  • 5.  
  • 6. T 3 : If A is a closed set and b is a point not in A, there exist disjoint open sets O A and O b containing A and b respectively. X
  • 7. T 4 : If A and B are disjoint closed sets in X, there exist disjoint open sets O A and O B containing A and B respectively X
  • 8. T 5 : If A and B are separated sets in X, there exist disjoint open sets O A and O B containing A and B respectively
  • 9. Hausdorff Spaces
    • All metric spaces are Hausdorff 3
    • The real number line with the usual topology is Hausdorff 4
      • Suppose we have an open set, S, of real numbers
      • For each p Є S we can find an ε >0, such that the ε -neighborhood of p forms an open set inside of S.
      • Suppose p, q Є S and are separated by a distance r. We can pick ε 1 and ε 2 for p and q , respectively, such that r/2 ≥ ε 1 > 0 and r/2 ≥ ε 2 > 0. Thus p and q will be separated by disjoint open sets.
  • 10. Subspaces of Hausdorff Spaces are also Hausdorff 5
    • If Y ⊂ X and (X, d) is Hausdorff, then any two distinct points x 1 , x 2 Є Y will also be in X. Since x 1 and x 2 are in X, there are disjoint open sets O 1 and O 2 containing x 1 and x 2 , respectively. Thus Y contains points that can be separated by disjoint open sets and is Hausdorff.
  • 11. Non-Hausdorff Spaces
    • Zariski Topology
    • Let X be [0, 1]∪[2, 3]. Define the following equivalence relation on X:
      • a ~ a + 2 for all 0 ≤ a ≤ 1 except for a = ½
      • b ~ b – 2 for 2 ≤ b ≤ 3 except for b = 5/2
    [0] [1] 5/2 1/2
  • 12. Proof:
    • Notice that every point in [0, 1] is identified with a point of [2, 3] except fpr the two points ½ and 5/2, which remain unidentified. Thus are contained in X/~. A set in X/~ is open iff it is open in X. However, we see that there are no disjoint open sets that contain ½ and 5/2. So X is Non-Hausdorff when considered under this equivalence relation.6
  • 13. Regular and Normal Spaces 7
    • Regular Space 8- A space which is both a T 1 and a T 3 space
    Normal Space 9 -A space which is both a T 1 and a T 4 space
  • 14.
    • Every Normal Space is Regular
    • Every Regular Space is Hausdorff
    • Not all Hausdorff Spaces are Regular
    • In the definitions of Normal and Regular we can replace the use of disjoint open sets, and use disjoint closures to yield the same spaces
    • The same is not true for the definition of Hausdorff
  • 15. Completely Hausdorff 10
    • T 2 ½ Axiom- If a and b are two points of a topological space X, there exist open sets O a and O b containing a and b, respectively, such that Ō a ∩ Ō b = Ø
    • A space that satisfies this axiom is called a completely Hausdorff space.
  • 16.  
  • 17. Compact Hausdorff Spaces
    • If a Hausdorff space is compact, then it is normal, and therefore also regular.
  • 18.
    • Let the sequence (S n ) in a Hausdorff Space have a limit, s. Suppose that the sequence has another limit, p, such that p≠s. Let U and V be disjoint neighborhoods of s and p, respectively. Then if we choose a sufficiently large n, S n Є U however, for a sufficiently large n, S n Є V. This contradicts the Hausdorff property because U ∩V ≠ Ø. Thus the sequence cannot have more than one limit.
  • 19. If X is Hausdorff and (S n ) is a sequence in X that converges to a point s Є X, and if y is an accumulation point of the set {S n | n = 1, 2, . . .}, then s = y.
  • 20.
    • Suppose s≠y. Then there exist open sets U, V Є X for s and y respectively such that U∩V = Ø. Also, since (S n ) converges to s, there exists a natural number, N such that n>N implies that S n Є U.
    • Let i be such that 1< i < N, and let W i , V i Є X be open sets for s i and y, respectively, such that W i ∩ V i = Ø, unless y= s i for some i.
    • Define V i ’= V i in case y≠s i , and V i ’= V in case y=s i and define V’ as the intersection of V with a finite collection of open sets, V i ’. So y Є V’ Є a system of neighborhoods around y.
  • 21.
    • Let z be such that z Є V’ and y≠z, then z≠s i for any i since if i > N, s i Є U and U∩V’ ⊆ U∩V = Ø,
    • And if i ≤ N, then either y=s i ≠z, or s i Є W i , and W i ∩V’ ⊆ W i ∩ V i = Ø.
    • Thus there is a neighborhood V’ of y, such that no point z ≠ y of the set {S n | n = 1, 2, . . .}, belongs to V’.
    • This contradicts the fact that y is an accumulation point of the set {S n | n = 1, 2, . . .},
    • thus s=y.
  • 22. Endnotes
    • Greever, John. Theory and Examples of Point-Set Topology. Claremont: Waybread Publications, 1990.
    • Steen, Lynn Arthur and J. Arthur Seebach, Jr. Counterexamples in Topology. New York: Dover Publications Inc., 1995.
    • Image of axiom spaces:
    • Sneddon, I. N. Ed. Andrew H. Wallace. “An Introduction To Algebraic Topology.” International Series of Monographs in Pure and Applied Mathematics. V.I. New York: Pergamon Press, 1957.
    • Baum, John D. Elements of Point Set Topology . New York: Dover Publications Inc., 1991
    • Sneddon, pg. 32.
    • Goodman, Sue E. Beginning Topology. Belmont: Brooks/Cole, 2005.
    • Baum, pg. 81.
    • Image-
    • Image-
    • Steen, pg. 13.
    • Steen, pg. 13.
    • Sneddon, pg. 31.