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.
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
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.
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.
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.
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: http://jtauber.com/2005/01/separation.png
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.
Be the first to comment