# Continuity in Topological Spaces ... M & M, Lemma 3.2 (ii) ... ...

#### Peter

##### Well-known member
MHB Site Helper
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

Last edited:

#### Opalg

##### MHB Oldtimer
Staff member
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 $$\displaystyle \overline{A} \subseteq f^{ -1} ( \overline{ f(A) } )$$ ... ...
This is actually a proof of (ii) $\Rightarrow$ (iii) in Lemma 3.2. So we are assuming that (ii) holds. In particular, since $\overline{ f(A) }$ is closed in $(Y,\tau')$, it follows from (ii) that $f^{ -1} ( \overline{ f(A) } )$ is closed in $(X,\tau)$. But $A$ is contained in the closed set $f^{ -1} ( \overline{ f(A) } )$, and therefore $\overline{A}$ (which is the smallest closed set containing $A$) is contained in $f^{ -1} ( \overline{ f(A) } )$.