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Definitions of Continuity in Topological Spaces ... Sutherland, Defintions 8.1 and 8.2 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
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 \(\displaystyle f\) is continuous iff it is continuous at every point of \(\displaystyle X\). ... ... "


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 \(\displaystyle U' \in T_Y\) where \(\displaystyle f(x) \in U'\) ... ...

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

Then \(\displaystyle x \in U\) ...

Also from Definition 8.1 we have \(\displaystyle U \in T_X\) ...

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

... that is Definition 8.2 holds at any \(\displaystyle x \in X\) ...


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

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

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

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

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

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

Therefore \(\displaystyle f^{-1} (V) \in T_X\) ... ...




Can someone please confirm that the above proof is correct ... and/or point out the shortcomings/errors ...



Hope someone can help ... ...

Peter


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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