Hausdorff Space and finite complement topology

In summary, the finite complement topology of the reals R is not Hausdorff because, by definition, any two points in R must have disjoint neighborhoods. However, in this topology, the complement of an open set is a finite set of elements, making it impossible for any two points to have disjoint neighborhoods. This is because the neighborhoods must be open sets, and in this topology, they can only be rays, not open intervals.
  • #1
Pippi
18
0
I want to come up with examples that finite complement topology of the reals R is not Hausdorff, because by definition, for each pair x1, x2 in R, x1 and x2 have some disjoint neighborhoods.

My thinking is as follows: finite complement topology of the reals R is a set that contains open sets of the form, (- inf, a1) U (a1, a2) U ... U (an, inf). The complement of an open set is the finite elements {a1, a2, ... an}. However, any two points I pick out of any open set have disjoint neighborhoods. How is it possible?
 
Physics news on Phys.org
  • #2
OK, pick two elements, how do they have disjoint neigborhoods?
 
  • #3
Ok, I see where my reasoning was wrong. The neighborhoods have to be open sets in this topology. If I pick two points, {0, 2}, in the open set (-inf, 1) U (1 inf), their neighborhoods can't be any open intervals and must be rays.

Cheers!
 

Related to Hausdorff Space and finite complement topology

1. What is a Hausdorff space?

A Hausdorff space is a topological space in which any two distinct points have disjoint neighborhoods. In other words, for any two points in the space, there exists open sets that contain one point and not the other.

2. How is a Hausdorff space different from other topological spaces?

Unlike other topological spaces, a Hausdorff space satisfies the Hausdorff axiom, which states that any two distinct points have disjoint neighborhoods. This axiom ensures that the space is separated and allows for more precise analysis of the space.

3. What is the finite complement topology?

The finite complement topology is a topology on a set where the open sets are the complements of finite sets. In other words, a set is open in this topology if its complement is finite or the entire set. This topology is commonly used in Hausdorff spaces.

4. How is the finite complement topology related to Hausdorff spaces?

The finite complement topology is commonly used in Hausdorff spaces because it satisfies the Hausdorff axiom. This topology allows for a more fine-grained analysis of the space and is useful in proving certain properties of Hausdorff spaces.

5. What are some examples of Hausdorff spaces?

Some common examples of Hausdorff spaces include Euclidean spaces (such as R^n), metric spaces, and topological manifolds. Other examples include the real line with the usual topology and the space of continuous functions on a compact interval with the uniform topology.

Similar threads

I Collection of finite unions of half-open intervals form an algebra
    • Topology and Analysis
Replies
3
Views
286
I How do I use the four axioms of a neighborhood to define an open set?
    • Topology and Analysis
Replies
2
Views
343
I How to define an open set using the four axioms of a neighborhood
    • Topology and Analysis
Replies
2
Views
374
I Fiber bundle homeomorphism with the fiber
    • Topology and Analysis
2
Replies
43
Views
1K
I On two definitions of locally compact
    • Topology and Analysis
Replies
5
Views
243
B Open sets in quotient topology
    • Topology and Analysis
Replies
8
Views
2K
I Limit Points & Closure in a Topological Space .... Singh, Theorem 1.3.7
    • Topology and Analysis
Replies
4
Views
2K
MHB Compact Topological Spaces .... Stromberg, Theorem 3.36 .... ....
    • Topology and Analysis
Replies
5
Views
1K
I Topology Words: Reasons for the particular names
    • Topology and Analysis
Replies
2
Views
1K
MHB Proving Topology in X: A Look at Union & Intersection
    • Topology and Analysis
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top