Homeomorphism and Connectedness

We give a QUICK intro to homeomorphisms and connected topological spaces. Then we show that connectedness is a topological property, meaning that it is preserved by homeomorphisms.

[thomasjefferson]

I like the idea that if a set is connected the only disjoint sets from the topology that form X are the entire space and the empty set.

Or in other words, the only two clopen sets that can form the topology is the space itself and the empty set. Otherwise its not connected. thanks viro.

I used this graph idea recently on this set of { (x, y in R^2 | x > 0 and y = sin(1/x)}. its graph is {x, f(x)} which is homeomprhic to the real line, and is thus path connected. i love this stuff.

you an also use this connected property to show if something isnt a homeomorphism.

for eample R -> S1 isnt a homeomorphism since if we take a point away from the line R we have created two connected componets. However if we remove a point from S1, its still connected! BLEW MY MIND, way easier than proving a homeomorphism typically.

[abdelazizmegdiche]

First ! Ever thought about doing a "tangent" video on category theory or topology on graphs ?