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Section Summary
Important ideas that we discussed in this section include the following.
A function \(f\) from a topological space \(X\) to a topological space \(Y\) is continuous if \(f^{-1}(O)\) is open in \(X\) whenever \(O\) is open in \(Y\text{.}\)
Two metric spaces \((X,d_X)\) and \((Y,d_Y)\) are metrically equivalent if there is a bijection \(f : X \to Y\) such that
\begin{align*}
d_X(x,y) \amp = d_Y(f(x),f(y))\\
d_Y(u,v) \amp = d_X(f^{-1}(u), f^{-1}(v))
\end{align*}
for all \(x,y \in X\) and \(u,v \in Y\text{.}\) That is, \(X\) and \(Y\) are metrically equivalent if there is a isometry \(f\) from \(X\) to \(Y\) such that \(f^{-1}\) is also an isometry. Topological equivalence is a less stringent condition. Two topological spaces \(X\) and \(Y\) are topologically equivalent if there is a continuous function \(f\) from \(X\) to \(Y\) such that \(f^{-1}\) is also continuous. That is, \(X\) and \(Y\) are topologically equivalent if there is a homeomorphism between \(X\) to \(Y\text{.}\)
A homeomorphism between topological spaces \(X\) and \(Y\) is a continuous function \(f\) from \(X\) to \(Y\) such that \(f^{-1}\) is also continuous. Two topological spaces \(X\) and \(Y\) are homeomorphic if there is a homeomorphism \(f : X \to Y\text{.}\)
A topological invariant is any property that topological space \(X\) has that must also be a property of any topological space homeomorphic to \(X\text{.}\) We can sometimes use topological invariants to determine if two topological spaces are not homeomorphic.
