Proving properties of maps between topological spaces

Question

I'm reading the section Maps between topological space from Isham CJ. Modern differential geometry for physicists. World Scientific; 1999.. Here he defines the map between two topological space $$ f:X \to Y $$ Induces a map from $P(X)$ to $P(Y)$, which is defined on a subset $A \subset X$ as $$ f(A) := \{\:f(x) \in Y |\: x \in A \:\} $$ and has the properties

$$ f(A \cup B) = f(A) \cup f(B) \\ f(A \cap B) \subset f(A) \cap f(B) $$

On the other hand, the inverse map from $P(Y)$ to $P(X)$ is $$ f^{-1}(A) = \{\: x \in X |\:f(x) \in A \:\} $$ This map has the following two properties $$ f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B) \\ f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B) $$

My objective is to start from the definition of the map and prove all of these 4 properties.

My take on the proof

Now, starting from the definition of the induced map I can prove the first property of the map \begin{align} f(A) &= \{\:f(x_1) \in Y |\: x_1 \in A \:\} \\ f(B) &= \{\:f(x_2) \in Y |\: x_2 \in B \:\} \\ \end{align} So, \begin{align} f(A\cup B) &= \{\:f(x_3) \in Y |\: x_3 \in A \cup B \:\} \end{align} Now, $x_3 \in A \cup B$ means $ x_3 \in A \text{ OR } x_3 \in B$. So, \begin{align} f(A\cup B) &= \{\:f(x_3) \in Y |\: x_3 \in A \text{ OR } x_3 \in B \:\} \\ \implies f(A\cup B) &= \{\:f(x_3) \in Y |\: x_3 \in A \:\} \cup \{\:f(x_3) \in Y |\: x_3 \in A \:\} \\&= f(A) \cup f(B) \end{align} Thats' how I prove the first property of the map. To prove the second property in the same spirit I do \begin{align} f(A\cap B) &= \{\:f(x_3) \in Y |\: x_3 \in A \cap B \:\} \\ \implies f(A\cap B) &= \{\:f(x_3) \in Y |\: x_3 \in A \text{ AND } x_3 \in B \:\} \\ \implies f(A \cap B) &= \{\:f(x_3) \in Y |\: x_3 \in A \:\} \cap \{\:f(x_3) \in Y |\: x_3 \in B \:\}\\ &= f(A) \cap f(B) \\ \end{align} Similarly I can prove the two properties for the inverse map.

I don't know how to get the subset relation for the second property. Another concern is that, can this kind of logic be used to prove the properties for the maps?

To prove subset relationships like $S\subset T$, it usually works to consider an element of $S$ and show it is also in $T$. — Adina Goldberg, Jul 28 '20 at 18:54
what is the map $P$ good for? I don't see how it contributes to anything. — Zest, Jul 28 '20 at 18:55
I see, so essentially the claim is $X\to Y$ induces a map $\mathcal{P}(X)\to \mathcal{P}(Y)$. — Zest, Jul 28 '20 at 18:57
The identity $f(A\cap B) = f(A)\cap f(B)$ is wrong in general and needs to be replaced by the weaker $f(A\cap B)\subset f(A)\cap f(B)$. — Louis Hainaut, Jul 28 '20 at 19:01
Note that at the end of your post you still "prove" this identity. This shows that reasoning with set equalities can be dangerous; unless the equality is obvious I suggest to prove such equalities by proving mutual inclusion. — Louis Hainaut, Jul 28 '20 at 19:09

score 0 · Answer 1 · answered Jul 28 '20 at 18:58

0

There is already a question about the intersection of preimages. In fact, the containment you write is an equality.

Also, yes, your technique should work to prove the other two statements as well.

answered Jul 28 '20 at 18:58

Adina Goldberg

454

Proving properties of maps between topological spaces

My take on the proof

1 Answers1