Definitions of Continuity in Topological Spaces ....

In summary, Definition 8.1 is essentially equivalent to Definition 8.2 iff f is continuous at every point of the domain of f.
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TL;DR Summary
I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ...
I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ...
I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ... see text below ...
I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ...

I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ...

I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...

Definitions 8.1 and 8.2 read as follows: ... ...
Sutherland - Defn 8.1 and Defn 8.2 ... .png

In the above text we read the following:

" ... ... Then one can prove that ##f## is continuous iff it is continuous at every point of ##X##. ... ... "I sketched out a proof of the above statement ... but am unsure of the correctness/validity of my proof ...

My sketch of the proof is as follows ...First assume f is continuous (Definition 8.1 holds true) ... ...

We are given (Definition 8.2) that ##U' \in T_Y## where ##f(x) \in U'## ... ...

Take ##U = f^{-1} (U')##

Then ##x \in U## ...

Also from Definition 8.1 we have ##U \in T_X## ...

and further ##f(U) = f(f^{-1}(U')) \subseteq U'## ...

... that is Definition 8.2 holds at any ##x \in X## ...Now assume that Definition 8.2 holds true at every x \in X ... that is f is continuous at every point x \in X ...

Let ##V \in T_Y## ... need to show ##f^{-1} (V) \in T_X## ... ...

Now ##x \in f^{-1} (V) \Longrightarrow f(x) \in V##

##\Longrightarrow## there exists a set ##U_x \in T_X## such that ##f(U_x) \subseteq V## by Definition 8.2 ...

But ##f(U_x) \subseteq V \Longrightarrow U_x \subseteq f^{-1} (V)##

Therefore for all ##x \in f^{-1} (V)## we have ##x \in U_x \subseteq f^{-1} (V)##

Therefore ##f^{-1} (V)## is open by Proposition 7.2 ...

Therefore ##f^{-1} (V) \in T_X## ... ...Can someone please confirm that the above proof is correct ... and/or point out the shortcomings/errors ... Hope someone can help ... ...

Peter====================================================================================The above post mentions Proposition 7.2 so I am providing text of the same together with the start of Chapter 7 in order to provide necessary context, definitions and notation ... as follows ... ...
Sutherland - 1 - Defn 7.1 and Propn 7.2 ... PART 1 ... .png

Sutherland - 2 - Defn 7.1 and Propn 7.2 ... PART 2 ... .png


Hope that helps ...

Peter
 
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Math Amateur said:
Summary:: I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ...
I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ...
I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ... see text below ...

I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ...

I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ...

I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...

Definitions 8.1 and 8.2 read as follows: ... ...View attachment 258280
In the above text we read the following:

" ... ... Then one can prove that ##f## is continuous iff it is continuous at every point of ##X##. ... ... "I sketched out a proof of the above statement ... but am unsure of the correctness/validity of my proof ...

My sketch of the proof is as follows ...First assume f is continuous (Definition 8.1 holds true) ... ...

We are given (Definition 8.2) that ##U' \in T_Y## where ##f(x) \in U'## ... ...

Take ##U = f^{-1} (U')##

Then ##x \in U## ...

Also from Definition 8.1 we have ##U \in T_X## ...

and further ##f(U) = f(f^{-1}(U')) \subseteq U'## ...

... that is Definition 8.2 holds at any ##x \in X## ...Now assume that Definition 8.2 holds true at every x \in X ... that is f is continuous at every point x \in X ...

Let ##V \in T_Y## ... need to show ##f^{-1} (V) \in T_X## ... ...

Now ##x \in f^{-1} (V) \Longrightarrow f(x) \in V##

##\Longrightarrow## there exists a set ##U_x \in T_X## such that ##f(U_x) \subseteq V## by Definition 8.2 ...

But ##f(U_x) \subseteq V \Longrightarrow U_x \subseteq f^{-1} (V)##

Therefore for all ##x \in f^{-1} (V)## we have ##x \in U_x \subseteq f^{-1} (V)##

Therefore ##f^{-1} (V)## is open by Proposition 7.2 ...

Therefore ##f^{-1} (V) \in T_X## ... ...Can someone please confirm that the above proof is correct ... and/or point out the shortcomings/errors ... Hope someone can help ... ...

Peter====================================================================================The above post mentions Proposition 7.2 so I am providing text of the same together with the start of Chapter 7 in order to provide necessary context, definitions and notation ... as follows ... ...View attachment 258281
View attachment 258282

Hope that helps ...

Peter

Looks correct to me.

Only remark I have is the following: At some point you consider ##U_x## without explicitely stating that it contains ##x##. So just add this small sentence to your proof.

Of course, I'm nitpicking but it doesn't hurt to be explicit.
 
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Thanks for your help, Math_QED ...

Peter
 
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1. What is the definition of continuity in topological spaces?

The definition of continuity in topological spaces is that a function f: X → Y between two topological spaces is continuous if the preimage of every open set in Y is an open set in X. In other words, for every open set U in Y, the set f-1(U) is open in X.

2. How does the definition of continuity in topological spaces differ from the definition in calculus?

In calculus, continuity is defined in terms of limits, where a function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. In topological spaces, continuity is defined in terms of open sets, where a function is continuous if the preimage of every open set is open.

3. Can you provide an example of a continuous function in topological spaces?

One example of a continuous function in topological spaces is the identity function f: R → R, where f(x) = x. This function is continuous because the preimage of any open set in R is an open set in R, and the open sets in R are defined by the standard topology.

4. How does the concept of continuity relate to the concept of connectedness in topological spaces?

In topological spaces, continuity is a necessary but not sufficient condition for connectedness. A function between two topological spaces must be continuous for the spaces to be considered connected, but there are cases where a function may be continuous but the spaces are not connected. For example, the function f: R → R defined by f(x) = 0 is continuous, but the spaces R and {0} are not connected.

5. Can continuity be defined in terms of closed sets in topological spaces?

Yes, continuity can also be defined in terms of closed sets in topological spaces. A function f: X → Y is continuous if the preimage of every closed set in Y is a closed set in X. This definition is equivalent to the one using open sets, as the preimage of a closed set is the complement of the preimage of the corresponding open set.

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