Defining Topological Spaces help

In summary, the conversation discusses the definition of a topology on the set of real numbers, with three possible collections of subsets being considered. The question asks which collection defines a topology, and whether this topology is connected and Hausdorff. The concept of subsets of the closed interval [0,1] is also mentioned. The response clarifies the interpretation of subsets of [0,1] and explains the criteria for a collection of subsets to form a topology.
  • #1
ikenmike05
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0

Homework Statement



Let ℝ be set of real numbers. Which of the following collection of subsets of ℝ defines a topology in ℝ.

a) The empty set and all sets which contain closed interval [0,1] as a subset.

b)R and all subsets of closed interval [0,1].

c)The empty set, ℝ and all sets such that A not subset of [0,1] and [0,1] not subset of A.

Determine if obtained topology is connected and Hausdorff.

The Attempt at a Solution



Im not sure how to interpret subsets of the closed interval [0,1] and this doesn't seem like it would be an open set.
 
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  • #2
"subset of [0, 1]" means exactly what it says- things like [1/2, 3/4], (1/3, 4/5), etc. This question does not ask it they are "open sets" in terms of the "usual topology", it asks whether the collection of all sets forms a topology.

You should recall that to be a "topology" a collection of subsets of set X must
1) contain X itself.
2) contain the empty set
3) contain the union of any sub-collection of these sets
4) contain the intersection of any finite sub-colection of these sets
 

What is a topological space?

A topological space is a mathematical concept that describes the structure of a set in terms of its open sets and their interactions. It is a foundational concept in topology, which is the study of the properties of geometric objects that are preserved under continuous transformations.

How is a topological space defined?

A topological space is defined by a set of points and a collection of open subsets of that set. The open subsets satisfy certain axioms, such as being closed under unions and finite intersections, that allow for the definition of continuity and other topological concepts.

What is the importance of defining topological spaces?

Defining topological spaces is important because it provides a way to study the properties of geometric objects in a more abstract and general way. It allows for the identification of common features and structures among seemingly different objects, making it a powerful tool in mathematics and other fields of science.

What are some examples of topological spaces?

Some common examples of topological spaces include the real line, the plane, and the sphere. Other examples include graphs, networks, and simplicial complexes. Topological spaces can also be defined on more abstract objects, such as sets of functions or groups.

How are topological spaces used in scientific research?

Topological spaces are used in a wide range of scientific research, including physics, biology, and computer science. They provide a framework for understanding the structure and behavior of complex systems, and can help identify patterns and relationships that may not be apparent at first glance. Topological spaces are also used in data analysis and machine learning to classify and analyze large datasets.

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