Summary

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 $Y .$
🔗

Two metric spaces $(X, d_{X})$ and $(Y, d_{Y})$ are metrically equivalent if there is a bijection $f : X \to Y$ such that

$\begin{aligned} d_{X} (x, y) & = d_{Y} (f (x), f (y)) \\ d_{Y} (u, v) & = d_{X} (f^{- 1} (u), f^{- 1} (v)) \end{aligned}$

for all $x, y \in X$ and $u, v \in Y .$ 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 $Y .$
🔗
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 $f : X \to Y .$
🔗
A topological invariant is any property that topological space $X$ has that must also be a property of any topological space homeomorphic to $X .$ We can sometimes use topological invariants to determine if two topological spaces are not homeomorphic.