Topology Q: Show f is Continuous in X with d and A

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In summary, the conversation discusses a question from the book "Introduction to Topology" and asks for a solution book recommendation. The question is about proving the continuity of a function defined on a metric space with a subset. The conversation also includes a discussion of using the triangular inequality to prove the continuity.
  • #1
mbarby
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hi all,
i am studying from croom's introduction to topology book. i came across such a question. and i don't have a clue as to how to start .
Let X be a metric space with metric d and A a non-empty subset of X. define f:X->IR by :

f(x): d(x,A), x E X (x is an element of X)
show that f is continuous.

also if you can point out a solution book for this book that would be rather nice, considering i am computer scientist studying the topic at home..
thx.
 
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  • #2
Hi mbarby! :smile:

You'll need to prove that

[tex]d(x,y)<\delta~\Rightarrow~|d(x,A)-d(y,A)|<\varepsilon[/tex]

Can you first prove that

[tex]-d(x,y)<d(x,A)-d(y,A)<d(x,y)[/tex]
 
  • #3
to prove that i use the triangular inequality
d(x,A) <= d(x,y)+d(y,A)
d(x,A) - d(y,A) <= d(x,y)

--->
-d(x,y) <= d(x,A) - d(y,A) <= d(x,y)

but honestly i couldn't connect it to any kind of a proof :/ ...
 
Last edited:
  • #4
Doesnt that immediately imply

[tex]|d(x,A)-d(y,A)|\leq d(x,y)[/tex]

and this would imply continuity...
 
  • #5


I would suggest approaching this problem by first reviewing the definition of continuity in topology. In general, a function f is continuous at a point a if for any open set U containing f(a), there exists an open set V containing a such that f(V) is a subset of U. This definition can also be extended to a function being continuous on a set A if it is continuous at every point in A.

In this specific problem, we are given a function f(x) = d(x,A), where d is the metric on X and A is a non-empty subset of X. To show that f is continuous, we need to show that it is continuous at every point in X. Let a be any arbitrary point in X and let U be an open set containing f(a). We want to find an open set V containing a such that f(V) is a subset of U.

Since A is a non-empty subset of X, there exists at least one point in A, say b, such that d(a,b) < d(a,A). This follows from the definition of distance between a point and a set. Now, consider the open ball B(b, r) centered at b with radius r = d(a,b) - d(a,A). By construction, this open ball contains a and is contained in A. Therefore, for any x in B(b,r), we have d(x,A) = d(x,b) < d(a,b) - d(a,A) + d(a,A) = d(a,b). This implies that f(x) < d(a,b) for all x in B(b,r).

Now, if we take V to be the open ball B(a,r) centered at a with radius r = d(a,b) - d(a,A), we have V contained in B(b,r) and therefore, f(V) is a subset of B(b,r), which is a subset of U. Hence, we have shown that for any point a in X, we can find an open set V containing a such that f(V) is a subset of U. This proves that f is continuous on X.

As for a solution book, I would recommend looking for solutions to specific problems online or consulting with a topology professor or fellow students for guidance. Good luck with your studies!
 

FAQ: Topology Q: Show f is Continuous in X with d and A

What is topology?

Topology is a branch of mathematics that studies the properties of geometric figures and spaces that are preserved under continuous deformations, such as stretching, twisting, and bending.

What is a continuous function?

A continuous function is a function where small changes in the input result in small changes in the output. In other words, if the input values are close together, the corresponding output values will also be close together.

What is the definition of continuity in topology?

In topology, a function f is continuous at a point x in a topological space X if for every open set V containing f(x), there exists an open set U containing x such that f(U) is a subset of V.

What is the role of the metric d in topology?

The metric d, also known as a distance function, is used to define the notion of distance between two points in a space. It is an essential tool in topology as it allows us to define open sets and ultimately, continuity of functions.

What does it mean for a function to be continuous on a set A?

A function f is continuous on a set A if it is continuous at every point in A. This means that the function behaves nicely on the set A, and small changes in the input values will result in small changes in the output values on A.

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