Definitions of Continuity in Topological Spaces ....

[Math Amateur]

TL;DR

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: ... ...

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 ... ...

Hope that helps ...

Peter

 

[member 587159]

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.

 

[Math Amateur]

Thanks for your help, Math_QED ...

Peter