Limit Points in a T_1 Space

In summary: Then, after you have established that every open neighborhood of x contains infinitely many points of A, you can use that fact to show that an infinite number of points in A are contained in any open neighborhood of x. Ultimately, it's important to make sure that your proof is logically sound and follows directly from the given information and definitions.
  • #1
Bashyboy
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Homework Statement



Problem: Let ##A## be an infinite subset of a ##T_1## space, and let ##x## be a limit point of ##A##. Prove that every open neighborhood of ##x## contains infinitely many points of ##A##.

Homework Equations

The Attempt at a Solution



First note that if ##\mathcal{O}## is an open neighborhood of ##x## not containing ##y##, then there exists open neighborhoods ##U## and ##V## containing ##x## and ##y##, respectively, but not containing the other point. Hence, ##\mathcal{O} \cap U \subseteq \mathcal{O}## is an open set containing ##x## but not containing ##y##.

Proof of main theorem: Let ##\mathcal{O}## be an open neighborhood of ##x##. Then there exists ##a_1 \in \mathcal{O} \cap (A-\{x\})##. By the observation made above, we can find an open neighborhood ##\mathcal{O}_2## of ##x## that doesn't contain ##a_1## and ##\mathcal{O}_1 \subseteq \mathcal{O}##. Hence, there exists ##a_2 \in \mathcal{O}_2 \cap (A-\{x,a_1\})##. Hence ##(a_n)## is an infinite sequence of distinct points in ##A## such that ##a_n## is in ##\mathcal{O}## for every ##n \in \mathbb{N}##, thereby proving that ##\mathcal{O}## contains an infinite number of points in ##A ~ \Box##.How does this sound? It seems little shaky; perhaps it could be made slightly more rigorous.
 
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  • #2
It sounds like you are assuming that there are an infinite number of points, and showing that there are an infinite number of subsets.
 
  • #3
I would recommend starting with the definitions of T_1 space, infinite subset, and limit point. Use your definitions as fuel for your proof.
 

1. What is a limit point in a T1 space?

A limit point in a T1 space is a point that can be approached by a sequence of distinct points in the space. This means that for every open neighborhood of the limit point, there exists at least one point in the sequence that lies within that neighborhood.

2. How is a limit point different from a boundary point in a T1 space?

A boundary point in a T1 space is a point that is both a limit point and an isolated point. This means that there exists an open neighborhood of the point that contains no other points in the space. In contrast, a limit point may have an infinite number of points in its neighborhood.

3. Can a T1 space have a finite number of limit points?

Yes, a T1 space can have a finite number of limit points. In fact, a discrete T1 space, where every point is isolated, has no limit points.

4. How are limit points related to accumulation points in a T1 space?

In a T1 space, limit points and accumulation points are equivalent. Both terms refer to points that can be approached by a sequence of distinct points in the space.

5. Are limit points unique in a T1 space?

Yes, limit points are unique in a T1 space. This means that for any given point in the space, there can only be one sequence of distinct points that converges to that point.

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