Topology Lecture 05 Supplemental: A bijective continuous function without continuous inverse

In this supplemental to lecture 05, we show an example of a continuous function that is bijective but lacks a continuous inverse.
This lecture follows Lee's "Introduction to topological manifolds", chapter 2.
A playlist with all the videos in this series can be found here:
• Topology

[@nektariosorfanoudakis2270]

For a less intuitive but simpler version we can take the identity function from a space to itself, but with two different topologies; the topology in the domain being strictly finer than the one in the codomain. For example X is a space with at least two points, the discrete topology in the domain, the trivial topology in the codomain. This example is the same, we can change the topology in the interval so that it becomes homeomorphic to the circle, but it won't be the subspace topology in R^2!

[@danielpfeffer2473]

One of the best explanations of the topology on the circle.

[@emilygeorgiadou7304]

This is an amazing explanation, also understood a lot better the definition of subspace topology! Thank you.

[@darrenpeck156]

Wonderful video. Thank you

[@yomo_13]

thnx
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