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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.

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  • $\begingroup$ For arbitrary topologies, we tend to check the pre-image of elements of some topological basis rather than arbitrary open sets. In maps between metric spaces, this gives us precisely the $\epsilon$-$\delta$ definition of continuity. $\endgroup$ – Omnomnomnom Oct 10 '15 at 12:24
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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....

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You don't always use the open-set-pullback argument to check for continuity. You need to use the machinery that's already been developed.

For your first example, note that $S^1$ has the subspace topology from $\mathbb{R}^2$, which in turn has product topology. Now what is the characterizing property for a product space?

As for the quotient space, look at how the (quotient) topology on $X/E$ is defined in the first place.

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