Continuity, and continuity in topology.
Metric spaces:
Neighborhood of a point $a$ is a Set of point $N$, such that $\exists\delta>0:B_\delta(a)\subset N$ ($B_r(x)$ = open ball at x of radius r)
Definition of open set: "A subset $O$ of a metric space is said to be open if it is a neighborhood to each of its points"
(It is now easy to go from this to the metrisable topological spaces)
Continuity:
($a$ is a point, $M$ and $N$ are neighborhoods (of the form $\forall M\exists N$), $M$ is a neigh. to $f(a)$, $N$ of $a$)
$\epsilon-\delta \iff f(B_\delta(a))\subset B_\epsilon(f(a))\iff B_\delta(a)\subset f^{-1}(B_\epsilon(a))\iff f(N)\subset M\iff N\subset f^{-1}(M)\iff f^{-1}(M)\text{ is a neigh. of }a$
This shows the "list" going from the well known $\forall\epsilon>0\exists\delta>0:|x-a|<\delta\implies|f(x)-f(a)|<\epsilon$
requiring only a leap of faith from $|\text{_}|$ to metrics (the ball definition) then it's straight on to the more topological $\forall N_{f(a)}, f^{-1}(N_{f(a)})\text{ is a neigh. of }a$
Topological space:
Given a topological space $(X,J)$ a set $N\subset X$ is a neighborhood of a point $a\in X$ if $N$ contains an open set that contains $a$
Definition of open set: "Thee members of $J$ are called 'open sets'"
The leap of faith ("Trust me, we're not just calling these 'open sets' rather than 'Thingymajig', think of them as open sets" and "Yeah, these are 'neighborhoods' we give them this name because they are like neighborhoods from metric spaces') required is really quite large.
I know it cannot be proved (there's no distance thus no open balls for the general topology) but I'd like some more discussion about accepting these definitions. From what I can tell.
"If we call an open set this, and a neighborhood that, later when we deal with distance and metrics, we can define open sets and neigh. using the metric, and show it has these properties, the that leads to continuity as you think of it"
So it's sort of going "from topology, to metric spaces" in terms of definitions and showing properties, however every book on the subject goes the other way. It builds on metric spaces then suddenly "we call these open sets!" I'd like to be a bit more confident and happy with the transition, recommended reading, examples, anything would be great.
general-topology analysis metric-spaces
Alec Teal
Alec TealAlec Teal
$\begingroup$ The notion of a topological space is more general than that of a metric space. Topologies that can be realized by a metric (distance function) are called metrizable. Some topologies are not metrizable, hence the greater generality of topological spaces compared to metric spaces. What precisely is your Question? $\endgroup$ – hardmath Aug 19 '14 at 0:13
When we say a metric topological space, we mean that opens sets in this topology is induced by the metric. So in this case definition for open sets is unique.
As for difference between open set and neighborhood, remark that each open set is a neighborhood for points inside it, and that each neighborhood contains an open neighborhood. So finally they are so "equivalent"(Of course not really, since a neighborhood is not always open) that usually we can interchange their names in most cases without making real mistakes
Petite EtincellePetite Etincelle
$\begingroup$ This doesn't answer the question at all. $\endgroup$ – Alec Teal Aug 16 '14 at 9:51
$\begingroup$ Maybe you could put your question a bit more clear? Because I read your post, then I think you are confused by two definitions in metric space and in topological space and the interchanged use of open set and neighborhood $\endgroup$ – Petite Etincelle Aug 16 '14 at 9:56
$\begingroup$ No, I'm happy with them, I just want to bridge the gap between them better. $\endgroup$ – Alec Teal Aug 16 '14 at 10:09
$\begingroup$ @AlecTeal There is literally no question mark in the post, so it's a little hard to evaluate whether this is answering the question or not. Anyway, you're right that it's more mathematically sensible to define open sets generally, and then look at special cases like metric spaces, but in practice we usually start with the simpler examples for pedagogical purposes. In this case, students often learn first $\mathbb{Q}$, then $\mathbb{R}$, then $\mathbb{R}^n$, then metric spaces, then topological spaces. Each is a generalization, each has open sets (which we first learn as "open intervals"). $\endgroup$ – Slade Aug 16 '14 at 14:53
$\begingroup$ @you-sir-33433 I know it's really poorly phrased. I think we first define topologies, We just call them "open sets" and ask they be a neighborhood to their points, then metric spaces are a special case, and we can satisfy the definitions using them. That's where I get a bit iffy, how do you show the topological definition implies the metric? $\endgroup$ – Alec Teal Aug 16 '14 at 14:57
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