Topology

>>8457774
>notice
novice

without a metric, continuity becomes a formal property that depends on the topology and just keeps track of open-closed data

but you should think of metric spaces as special cases of topological spaces not the other way around

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Watch this meme lecture https://www.youtube.com/watch?v=7G4SqIboeig

Continuity <-> preimage of open sets are open

>>8457812
pretty pointless to put an <-> there when its the definition

>>8457816
:= <-> <->

>>8457774

You can prove that in a metric space: epsilon-delta continuity <---> preimage of open is open

Now the epsilon-delta definition does not generalize to arbitrary topologies, but the preimage of open is open definition does.

>>8457816
thata not the pointwise definition tho, nigger :)

>>8457849
>the pointwise definition
which one is that? you mean the sequential one which is just equivalent?

>>8457816
No it's not pointless. I always do it too instead of "Def" or similar sheit.

>>8457851
nope
>which is just equivalent
ha, so the <=> isn't that pointless then, right?

>>8457853
imo if you say continuity <-> preimage of open sets are open it makes it seem like continuity is defined as something other than that

>>8457857
what? your post seems confused. what is the pointwise definition you're talking about?

>>8457774
>Why does the generalization of topology in metric spaces to topological spaces make sense?
It makes sense because the definition of a topological space takes results from a metric spaces as a definition.
In particular, the finite intersection of open sets is open, the arbitrary union of opens sets is open, and that the whole set and the empty set are (cl)open.
In fact, elements of the topology are called open sets.

What do you know about topology so far? It'd be easier to motivate it based on your answer.

Also you don't need to know analysis to study topology, but you will need it later down the line if you study metric spaces (which should come before topology).

>>8457816
All definitions are iff anyway, so it doesn't matter.

>>8457812
>1 hour 17 minutes and 5 seconds
The lecture is way too slow and a waste of time.