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

#### Peter

##### Well-known member
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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 $$\displaystyle f$$ 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 ... ...

Hope that helps ...

Peter