Through comparing topologies, one could deduce a more compact notation for continuity. It also leads to a nice description of continuous bijections and homemorphisms.

Comparison of topologies and continuity

Given two topologies \(\tau_1, \tau_2\) of \(X\). We say that \(\tau_1\) is coarser than \(\tau_2\) iff \(\tau_1 \subseteq \tau_2\). We also say that \(\tau_1\) is finer than \(\tau_2\) iff \(\tau_2 \subset \tau_1\). Intuitively a topology is coarser than another if it has more open sets.

This has links with continuity as demonstrated below.

Theorem: \(\text{id}_X: (X, \tau_1) \to (X, \tau_2)\) is continuous iff \(\tau_2 \subseteq\tau_1\)

As such we could see that \(\text{id}_X: (X, \tau_1) \to (X, \tau_2)\) is continuous iff \(\tau_2\) is coarser than \(\tau_1\).

This idea generalises to arbitrary functions. Note that for every function \(f: X \to Y\), it induces a function \(f': P(X) \to P(Y)\), where \(P(X), P(Y)\) are the power sets of \(X\) and \(Y\) such that \(f'(\Omega) := \bigcup_{S \in \Omega}\text{Im}_f(S)\) for all \(\Omega \in P(X)\). Depending on context, we shall use \(f'\) and \(f\) interchangebaly.

For full generality we see that

Theorem: \(f: (X, \tau_1) \to (X, \tau_2)\) is continuous iff \(f^{-1}(\tau_2) \subseteq \tau_1\)

Theorem: \(f: (X, \tau_X) \to (Y, \tau_Y)\) is continuous iff \(f^{-1}(\tau_Y) \subseteq \tau_X\)

This is just a fancy way of saying that \(\; \forall U \in \tau_Y, f^{-1}(U) \in \tau_X\), the classical definition of continuity.

In fact, given some map \(f: X \to Y\) where \(Y\) has some topology \(\tau_Y\). We could define a topology on \(X\), \(\tau_X\), as follows

\[\tau_x := f^{-1}(\tau_Y) = \{ U \subset X \; \vert \; \exists V \in Y \text{ s.t. } f^{-1}(V) = U\}\]

This makes \(f: (X, \tau_X) \to (Y, \tau_Y)\) continuous. It’s also the coarsest topology on \(X\) such that \(f\) is continuous. If \(X \subset Y\) and \(f\) is the inclusion map, then we call \(\tau_X\) the subspace topology.

Bijections and Homeomorphisms

If \(f\) is a bijection, we have \(f^{-1}\) well defined and \((f^{-1})^{-1}=f\). As such,

Theorem: Let \(f: (X, \tau_X) \to (Y, \tau_Y)\) be a bijection. Then \(f\) is continuous iff \(f^{-1}(\tau_Y) \subseteq \tau_X\) iff \(\tau_Y \subseteq f(\tau_X)\)

It follows that

Theorem: Let \(f: (X, \tau_X) \to (Y, \tau_Y)\) be a continuous bijection with a continuous inverse, then \(f(\tau_X) = \tau_Y\)

Proof: As \(f^{-1}: Y \to X\) continuous, from the previous theorem we have \((f^{-1})^{-1} (\tau_x) \subseteq \tau_Y\) so \(f(\tau_X) \subseteq \tau_Y\). Similarly as \(f\) is continuous we have \(\tau_Y \subseteq f(\tau_X)\). Result follows by double inclusion. \(\square\)

This suggests that homemorphisms preserve all topological properties.

Further reading

A lot of these ideas extend to describing topology using category theory. One could refer to this book which goes further and introduces a number of very useful characterisations of the subspace, quotient and product topology.

Thanks to Joshua Lau for providing helpful comments.

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