What is the equivalence of definitions for completely regular spaces?

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In summary, there are two definitions of "completely regular" - one from the class and online, and the other from the textbook. The definitions involve a space being completely regular if for every closed subset and point in the space, there is a continuous function with specific properties. Despite the seemingly different definitions, it is possible to show that these two definitions are equivalent.
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Cincinnatus
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I have seen two different definitions of "completely regular" (one in my class and online, and the other in my textbook). I am having trouble seeing how these definitions are equivalent.

A space S is said to be completely regular if for every closed subset C of S and every point x in S-C there is a continuous function from S to I such that
f(x)=0 and f(C)=1

A space S is said to be completely regular if for every point p of S and for every open set U containing p, there is a continuous function of S into I such that f(p)=0 and f(x)=1 for all points x in S-U.
 
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Let C be the complement of U?
 
  • #3
Ohhhh

I feel stupid now

I was thinking about this late at night, I'm not usually this dense.
 
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1. What is a completely regular space?

A completely regular space is a topological space in which every point has a neighborhood that can be separated from any closed set by a continuous function. This means that for any point in the space and any closed set not containing that point, there exists a continuous function that assigns values in the real numbers such that the point is mapped to a value less than the closed set and all other points are mapped to values greater than the closed set.

2. How is a completely regular space different from a regular space?

A completely regular space is a stronger condition than regularity. While regularity only requires points and closed sets to be separated by open sets, completely regular spaces require separation by continuous functions. This means that all completely regular spaces are also regular, but not all regular spaces are completely regular.

3. What are some examples of completely regular spaces?

Some examples of completely regular spaces include metric spaces, topological groups, and compact Hausdorff spaces. In fact, all Tychonoff spaces (topological spaces where all finite products of closed sets are closed) are completely regular.

4. Are all completely regular spaces also normal?

Yes, all completely regular spaces are also normal. This means that in addition to being able to separate points and closed sets, completely regular spaces can also separate disjoint closed sets. This is because in completely regular spaces, closed sets can be separated by continuous functions, which can be used to construct a partition of unity that separates the disjoint closed sets.

5. What is the importance of completely regular spaces in mathematics?

Completely regular spaces are important in various areas of mathematics, including functional analysis, topology, and algebraic geometry. They provide a natural setting for studying continuous functions and have many useful properties, such as being locally compact and having a Stone-Čech compactification. Completely regular spaces also play a significant role in the development of the theory of topological vector spaces and topological algebras.

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