Is 1 in the Closure of (2,3] in the Standard Topology on the Real Numbers?

In summary, the conversation discusses a problem in topology where the goal is to show that 1 is not an element of the closure of the interval (2,3] in the topological space of real numbers with the standard topology induced by the Euclidean metric. The conversation also clarifies some notations and definitions, such as U for the open topology and Cl for the closed space. The main focus is on understanding the definition of closure of a set and how it applies to this problem.
  • #1
SYoungblood
64
1

Homework Statement



Hello All, I am experiencing Adventures in Topology. So far, so good, but I have an issue here.

In the topological space (Real #s, U), show that 1 is not an element of Cl((2,3]).

Homework Equations



The closed subsets of our topological space are the converses of the given set. Over the set of real numbers, in an open topology, we have a converse of (-inf, 2] u (3,inf)

The Attempt at a Solution


[/B]
By that definition, 1 is most certainly an element of Cl((2,3]), and I am simply not seeing otherwise. Any thoughts?

Thank you,

SY
 
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  • #2
What do you mean by #s and U? What exactly are your open sets, or likewise closed sets? Does Cl mean closure? In addition I assume by converse you mean the complement, but I'm not sure.

How is the closure of a set defined?
 
  • #3
fresh_42 said:
What do you mean by #s and U? What exactly are your open sets, or likewise closed sets? Does Cl mean closure? In addition I assume by converse you mean the complement, but I'm not sure.

How is the closure of a set defined?

Real #s is the best notation I could think of for the set of Real Numbers. The set (2,3] is neither open nor closed under open topology, which I noted as U, in the fashion of my test. Cl is a closed space. As an example, in the space (Real #s, U), Cl((0,1)) = [0,1], the compliment of the interval (0,1) over the set of real numbers.

With that in mind, again, I am still not seeing how to exclude 1 from the interval of Cl ((2,3]).
 
  • #4
To get the wording straight. You consider the topological space ##\mathbb{R}## with the standard topology, which is induced by the Euclidean metric, which measures distances as ##d(x,y) = |x-y|##. Now ##1 \notin (2,3]## and you want to show, whether ##1## can be in the closure of this interval or not. Here is the essential part of your conclusion, because it requires a definition of the closure of a set. This can be done in several ways, purely topological or by means of the metric. But you have said, that ##Cl((2,3])=[2,3]## according to your example. So where is the problem?

Otherwise, you have to say, which definition of a closure of a set you want to use, before we can achieve a formal proof. What does your book say?
 

1. What is a topological space?

A topological space is a mathematical concept used to describe the properties and relationships between points in a space. It is a set of points, along with a set of rules (called the topology) that determine which collections of points are considered "near" each other.

2. What is the difference between a topological space and a metric space?

A metric space is a type of topological space where the distance between points is defined by a metric (a function that assigns a distance value to pairs of points). A topological space, on the other hand, only defines a notion of "nearness" without specifying any specific distances between points. In other words, a metric space is a more specific type of topological space.

3. How is a topological space represented mathematically?

A topological space is represented as a pair (X, T), where X is the set of points and T is the topology on X. The topology is a collection of subsets of X that satisfy certain properties, such as being closed under intersection and union. This representation allows for the analysis of the topological properties of a space.

4. What is the importance of topological spaces in mathematics?

Topological spaces are important in mathematics because they provide a framework for studying and understanding abstract concepts such as continuity, convergence, and connectedness. They are used in various branches of mathematics, including topology, analysis, and geometry.

5. How are topological spaces used in real-world applications?

Topological spaces have many applications in real-world problems, such as in computer science, physics, and engineering. They can be used to model networks, study the behavior of systems, and analyze data. For example, topological data analysis is a growing field that uses topological techniques to analyze complex data sets.

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