## Showing that two mappings are continuous. Introduction to topology

I have a course were we are learning about topology and have to show in two different exercises that two different mappings are continuous. I think I have a solution. However, as these concepts a new to me, I would like to ask if this is indeed correct and if there is a better way to solve them.

The first problem: Let $$(X,d_X)$$ and $$(Y,d_Y)$$ be metric spaces and define $$d_{X\times Y}:(X\times Y)\times (X\times Y) \rightarrow \mathbb{R}^+_0$$ by

$$d_{X\times Y}((x_1,y_1),(x_2,y_2))=max(d_X(x_1,x_2),d_Y(y_1,y_2))$$

Then show the projection

$$p_X:X\times Y \rightarrow X, p_X(x,y)=x$$

is continuous.

I would then solve this by showing that for every open set V in X then $$p_X^{-1}(V)$$ is an open set in $$X\times Y$$. So in mathematical terms

$$V \subseteq X$$ and

$$\forall x\in V, \exists \delta_V >0 : B^X_{\delta_V}(x)\subseteq V$$ and $$B^X_{\delta_V}(x)=\{y\in X|d_X(x,y)<\delta_V\}$$

We then need there to exist a $$\delta_{X\times Y}$$ so that $$\forall (x,y)\in p_X^{-1}(V), \exists \delta_{X\times Y} >0 : B^{X\times Y}_{\delta_{X\times Y}}((x,y))\subseteq p_X^{-1}(V)$$

Where $$p_X^{-1}(V)=V\times Y$$ as the second coordinate in $$(x,y)$$ is just dropped and can therefore be anything.

As every $$x_1 \in V$$ has a value $$\delta_X$$ we can now look at every $$(x_1,y)\in p_X^{-1}(V)$$, where it is the same element $$x_1$$ as before, then $$B^{X\times Y}_{\delta_{X\times Y}}((x_1,y))\subseteq p_X^{-1}(V)$$ if $$0<\delta_{X\times Y} \leq \delta_X$$.

I conclude this as for $$B^{X\times Y}_{\delta_{X\times Y}}((x,y))\nsubseteq p_X^{-1}(V)$$ then there has to be a point $$(x_2,y_2)$$ where $$d_X(x_1,x_2)<\delta_{X\times Y}\leq\delta_X$$ and $$x_2\notin V$$ as every possible $$y_2$$ is in $$p_X^{-1}(V)=V\times Y$$. But this is contradictory to what we know as $$\delta_X$$ is chosen so every $$x_2$$ where $$d_X(x1,x2)<\delta_X$$ is in V.

We can therefore conclude that for every open subset of X, called $$V$$, then $$p_X^{-1}(V)$$ is an open subset of $$X \times Y$$ as $$\forall (x,y)\in p_X^{-1}(V), \exists \delta_{X\times Y} >0 : B^{X\times Y}_{\delta_{X\times Y}}((x,y))\subseteq p_X^{-1}(V)$$ is fulfilled and the projection is therefore continuous.

The second problem is so similar that it is not currently added.

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