Studying some product topology, but I have a problem in my notes which is left as an exercise to think about, but I am not sure how to solve it. Help with his problem will really help me understand continuity of products on topology much better.
Problem: How can one prove that $$\Psi(g,h)=g(0)-h(0):C[0,1]_{\text{point convergence}}\times C[0,1]_{\text{point convergence}}\rightarrow \mathbb{R}_{(-)}$$ where $\mathbb{R}_{(-)}$ is the usual topology, $C[0,1]_{\text{point convergence}}$ is the pointwise convergence topology.
I have a further question which results from this question, and that is, I think that $\{(g,h):g(0)\ge h(0)\}$ is a closed subeset of $C[0,1]_{\text{point convergence}}\times C[0,1]_{\text{point convergence}}.$ However it frustrates me because I think I still lack alot of basic techniques and methods to show this is true. I mean I can see why this would be true, but I can formally prove it.
Can anyone please help me? I am still in the process of getting used to solving problems in topology. Seeing how to solve problems like these will help me greatly.
I'm assuming that the problem consists in proving that $\Psi$ is continuous. It will be enough to prove that the function $\varphi\colon C\bigl([0,1]\bigr)\longrightarrow\mathbb R$ defined by $\varphi(g)=0$ is continuous. This is so beause if $O$ is an open subset of $\mathbb R$, then$$\varphi^{-1}(O)=\left\{f\in C\bigl([0,1]\bigr)\,\middle|\,f(0)\in O\right\},$$which is an open subset of $C\bigl([0,1]\bigr)$, by the definition of the topology of pointwise convergence.
Now, since $\Psi(g,h)=\varphi(g)-\varphi(h)$, $\Psi$ is continuous.
Can you use this approach to answer the other question?