Hausdorff Spaces: Does Convergence to One Point Characterize?

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In summary, the conversation discusses whether every sequence converging to at most one point characterizes Hausdorff spaces. It is shown that this is not the case, using the example of an uncountable set with the cocountable topology. Additionally, it is proven that in this topology, limits of convergent sequences must be unique.
  • #1
R136a1
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Hello everybody!

It is known that in Hausdorff spaces that every sequence converges to at most one point. I'm curious whether this characterizes Hausdorff spaces. If in a space, every sequence converges to at most one point, can one deduce Hausdorff?

Thanks in advance!
 
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  • #2
Not necessarily. Let ##X## be an uncountable set with the cocountable topology ##\mathcal{T} = \{U\subseteq X:X\setminus U \text{ is countable}\}##. Assume there exist two distinct points ##p,p'## and two neighborhoods ##U,U'## of the two points respectively such that ##U\cap U' = \varnothing ##. Then ##X\setminus (U\cap U') = (X\setminus U)\cup (X\setminus U') = X##. But ##(X\setminus U)\cup (X\setminus U')## is a finite union of countable sets which is countable whereas ##X## is uncountable thus we have a contradiction. Hence ##X## is not Hausdorff under the cocountable topology.

Now let ##(x_i)## be a sequence in ##X## that converges to ##x \in X## and let ##S = \{x_i:x_i\neq x\}##. This set is countable so ##U\setminus S## must be a neighborhood of ##x## in ##X##. Thus there exists some ##n\in \mathbb{N}## such that ##i\geq n\Rightarrow x_i\in U## but the only distinct element of the sequence that is in ##U## is ##x## so ##x_i = x## for all ##i\geq n## i.e. any convergent sequence in ##X## must be eventually constant under the cocountable topology. Hence limits of convergent sequences must be unique (the map prescribing the sequence must be well-defined).
 
  • #3
Great! Thanks a lot! The example you gave is very interesting since it has the same convergent sequences as the discrete topology.
 

1. What is a Hausdorff space?

A Hausdorff space is a type of topological space that satisfies a separation axiom known as the Hausdorff condition. This means that for any two distinct points in the space, there exist open sets that contain one point but not the other.

2. What does it mean for a space to converge to one point?

Convergence to one point in a Hausdorff space means that as the elements of a sequence of points get closer and closer to a certain point, they eventually all converge to that point. This is also known as pointwise convergence.

3. How is convergence to one point related to the Hausdorff condition?

In a Hausdorff space, convergence to one point is a necessary and sufficient condition for a sequence of points to converge. This means that if a sequence of points converges in a Hausdorff space, it must converge to a single point, and conversely, if it converges to a single point, it must converge in the space.

4. Are all Hausdorff spaces also spaces that converge to one point?

Yes, all Hausdorff spaces also have the property of convergence to one point. However, not all spaces that have convergence to one point are necessarily Hausdorff spaces. There are other types of topological spaces that can also have this property.

5. What is the significance of convergence to one point in mathematics?

Convergence to one point is an important concept in mathematics, particularly in the study of topological spaces. It allows for the definition of limits, continuity, and compactness, which are fundamental concepts in analysis and other areas of mathematics. In addition, the property of convergence to one point has applications in fields such as physics, engineering, and computer science.

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