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I am reading Aisling McCluskey and Brian McMaster: Undergraduate Topology, Oxford University Press, 2014... ... and am currently focused on Chapter 3: Continuity and Convergence ...

I need help in order to fully understand M & M's proof of Lemma 3.2 (ii) ...

Lemma 3.2 (plus the definition of continuity) reads as follows:

The proof of Lemma 3.2 (ii) reads as follows:

In the above proof we read the following:

" ... ... But \(\displaystyle A \subseteq f^{ -1} ( f(A) ) \subseteq f^{ -1} ( \overline{ f(A) } )\) ... ...

... ... therefore \(\displaystyle \overline{A} \subseteq f^{ -1} ( \overline{ f(A) } )\) ... ... "

Can someone please demonstrate formally and rigorously that \(\displaystyle A \subseteq f^{ -1} ( f(A) ) \subseteq f^{ -1} ( \overline{ f(A) } )\) implies that \(\displaystyle \overline{A} \subseteq f^{ -1} ( \overline{ f(A) } )\) ... ...

Help will be much appreciated ...

Peter

I need help in order to fully understand M & M's proof of Lemma 3.2 (ii) ...

Lemma 3.2 (plus the definition of continuity) reads as follows:

The proof of Lemma 3.2 (ii) reads as follows:

In the above proof we read the following:

" ... ... But \(\displaystyle A \subseteq f^{ -1} ( f(A) ) \subseteq f^{ -1} ( \overline{ f(A) } )\) ... ...

... ... therefore \(\displaystyle \overline{A} \subseteq f^{ -1} ( \overline{ f(A) } )\) ... ... "

Can someone please demonstrate formally and rigorously that \(\displaystyle A \subseteq f^{ -1} ( f(A) ) \subseteq f^{ -1} ( \overline{ f(A) } )\) implies that \(\displaystyle \overline{A} \subseteq f^{ -1} ( \overline{ f(A) } )\) ... ...

Help will be much appreciated ...

Peter

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