I am having some troubles in proving continuity of functions between topological spaces. I'm still working on some basic examples. I understand that sums, products and composition of continuous functions are also continuous. For example, consider the function $$f: \mathbb{R}\rightarrow S^1$$ given by $$f(t)=(\cos t,\sin t).$$ How do I prove continuity using the definition? I can pick one open set in $S^1$ with the usual topology but I don't know how to compute its pre-image. Another example is the quotient map $$q:X\rightarrow X/E$$ given by $$q(x)=[x]_E,$$ where $E$ is as equivalence relation. My problem is the same, I don't know how to compute the pre-image of an arbitrary open set. I'd really appreciate if someone could give me some help.
[Math] Continuous functions between topological spaces
general-topology
Best Answer
For your first example, let $V\subset \mathbb S^1$ be an open set. Then by the definition of subspace topology, there is an open set $U\subset \mathbb R^2$ so that $V = U\cap \mathbb S^1$. Now using the fact that $g(t) = (\cos t, \sin t)$ is a continuous function from $\mathbb R$ to $\mathbb R^2$, $g^{-1}(U)$ is open in $\mathbb R$. But $$f^{-1}(V) = f^{-1}(U\cap \mathbb S^1) = g^{-1}(U) ,$$ thus $f^{-1}(V)$ is open and so $f$ is continuous.
Now for the quotient map $q : X\to X/E$, continuity of $q$ follows directly from the definition of the topology on $X/E$. We say that a set $U \subset X/E$ is open if and only if $q^{-1}(U)$ is open in $X$. Thus by definition $q$ is continuous....