Is it even possible? Or am I just flat out wrong once again?

[rudinreader]

Is it even possible? Or am I just flat out wrong once again?

Whenever I really need to procrastinate, I turn to writing a calculus book. In this case.. For about three days I pursued the problem of whether or not you can teach topology before "rigorous Calculus".

Is it even possible, to teach topology before rigorous Calculus (i.e. the Calculus that requires proof but is usually not yet called "Analysis")

In the end, the following two paragraphs is pretty much the fruit of my 3 days of effort, and is probably either a) wrong, or b) written somewhere else. Writing a book is hard!

A Topology on a Set

We now pursue another "notational convention" that allows us to mathematically discuss an otherwise difficult to express intuitive idea. A "topology" on a set has a lot of very important mathematical ideas associated with it, one of which is the intuitive idea of "connectedness".

Let G be any set. A topology on G is a collection τ of subsets (not points!) of G called regions, which must satisfy the following:

1) G itself is a "region" (G ∈ τ), and {} is a "region" ({} ∈ τ)

2) If U is any union of regions then U itself is also a "region" (U = ∪ R_α ⇒ U ∈ τ)

3) If U is a finite intersection of regions the U itself is also a "region" (U = R_1 ∩ R_2 ⇒ U ∈ τ)


In other words, a "topology" on a set identifies "regions" from the point of view of the game "Axis and Allies". At any point in the game, your "opponents region" may be consist of any union or (finite) intersection of "regions" on the game board. To win the game, you try to end up with the region G (the whole space), and to lose the game, you end up with the region {}.


A Finite (Geography) Example

Let X be the space known as the "greater HI-WA-AK area", where there is only one "route" R between Washington and Alaska (the "WA-AK ferry"), and no routes in or out of Hawaii. We define X and its topology as follows:

X = {HI,WA,AK,R}
τ is the unions and intersections (taken in either order) of S = {{HI},{WA},{AK},{WA,AK,R},X}.

Since R is just a "route" (the "ferry"), we don't consider it as a bonifide region. But nonetheless we must consider R as a part (point) of the space X (otherwise there's no information to suggest mathematically that WA-AK are connected to each other). The "subregion" known as the "greater WA-AK area" is defined as {WA,AK,R}, which evidently should also contain this "infastructure" R. After taking a union of {WA},{AK}, we get that {WA,AK} is theoretically also a region (indeed a disconnected region), which in terms of "Axis and Allies" is the region obtained from {WA,AK,R} after the ferry is bombed/destroyed.


In the sections that follow, it will be described why {WA,AK,R} is a "connected region", while {HI,WA}, {HI,AK}, and X itself are "disconnected". The key mathematical observation is that a region with two or more subregions is disconnected (has distinct regions with no "route" to one another) if and only if the complement of a subregion is also a subregion...

 

[matticus]

i would keep reading it. i liked the axis and allies analogy.

 

[rudinreader]

Actually with the above as a starting point, I think I see a way how to do it (present abstract topology before "rigorous Calculus") - and preferably quickly. The point is that probably it is best to just get a taste, and not go deep. Not even mention countability/uncountability, or perhaps return to that after some heaps of Calculus..

I am wondering, is the set theory too hindering for those who haven't studied proofs before? Or can a beginner get aquainted with the notation well enough as long as there are some concrete examples?

Anyways, here's a continuation of the train of thought...

Going forward - the definition of topology given above will be referred to as "Definition 1.0".

Constructing Topologies on a Space X

For a space of more than say 10 points, there are a ton of topologies that can be defined on X besides the smallest and the largest one.

The "smallest" and "largest" topologies on a space X are respectively the "trivial topology" http://en.wikipedia.org/wiki/Trivial_topology, and the "discrete topology" http://en.wikipedia.org/wiki/Discrete_space. In the trivial topology, τ = {X,{}}. X is the only (nonempty) region, and it is a connected space (again we will define "connected" later). In the discrete topology, τ = P(X), the set of all subsets of X. Every singleton {p} of X is a region in the discrete topology and it is "completely/everywhere disconnected".

We can construct other (usually more interesting) topologies, as hinted in the "geography" example, as follows.

Definition 1.1: If S is any collection of subsets of X, we define the topology generated by S as the "smallest topology containing S". In terms of sets, τ is the intersection of all topologies that contain S. Since all topologies contain {{},X}, and P(X) always contains S, this definition implies {{},X} ⊆ τ ⊆ P(X), so that τ is not empty. And τ immediately satisfies 1,2,3 of the definition of a topology (exercise), so that the term "smallest topology containing S" makes sense. Sometimes this operation is also called "the closure with respect to unions and finite intersections".

Remark: if you add the whole space X to S you get the same topology (exercise). So we will always assume X is in S or otherwise the union of sets in S is X. As such S is usually called a subbasis for the topology generated by S.

Having constructed a topology from a subbasis S, an important practical problem is then being able to identify which sets are regions, and which are not. We again look at the geography example:

X = {HI,WA,AK,R}, S = {{HI},{WA},{AK},{WA,AK,R},X}.

Is was discussed above, we don't want {R} to be a region (in the generated topology), it's just a "connection" between Washington and Alaska. But how can we tell by inspection that {R} is not a region? In this case the finite intersections of S turn out to be S itself. So we can deduce that C = {R} is not a region, because there is no "base" U such that R ∈ U ⊆ C. Or equivalently, every "base" that contains R is not contained in C. "base" is somewhat synonymous for "region", but we want to distinguish the terms because a region (in the topology) is built up from bases (from a subbasis - or later on, a basis).

Definition 1.2: Let S be a subbasis for X. A finite intersection U of sets in S is called a base.

Theorem 1.3: A subset U of X is a region if and only if one of the following (equivalent statements) is true:

1) U is a union of bases.
2) For every x in U there exists a base V such that x ∈ V ⊆ U.

proof:

That 1),2) are equivalent is an important exercise in set theory (for the reader). Otherwise this proof requires significant set theory notation (translation exercise). Suppose U_α = ∪_β(α) R_αβ(α) is a union of bases for all α. Then evidently U = ∪_α(∪_β(α) R_αβ(α)) = ∪_α,β(α) R_αβ(α) is also a union of bases. Now suppose U_1 = ∪_β(1) R_β(1), U_2 = ∪_β(2) R_β(2) are unions of bases. Then evidently, U_1 ∩ U_2 = (∪_β(1) R_β(1)) ∩ (∪_β(2) R_β(2)) = ∪_β(1),β(2) (R_β(1) ∩ R_β(2)), so that U is also a union of bases. It follows that the set "T" of all unions of bases (implicitly along with X and {}) is a topology that contains the subbasis, since each set in the subbasis is a finite intersection of itself. But since every topology must contain unions of bases (why?) it follows that T is precisely the topology generated by the subbasis. The theorem now follows, because a set U is a region if and only if it is a union of bases. (The case U = X is dealt with by assuming the union of the subbasis is X, or otherwise X is in S. The case U = {} is "assumed vacuous".)


We conclude with the definition of a basis.

Definition 1.4: Suppose B is a collection of subsets who's union is X, such that B satisfies the following:
(***) for any B_1,B_2 in B and for all x in B_1 ∩ B_2 there exists a U in B such that x ∈ U ⊆ B_1 ∩ B_2.
Then the proof of 1.3 can be modified to show, that the set of all unions of sets in B is a topology in X. Then B is called a basis for this topology. And in fact, this is precisely the topology it generates.

We have concentrated instead on the subbasis so far because it is the most intuitive from the view of not placing any restriction on which subsets to include in the topology. However, since the subbasis S in the geography example is closed under intersection, it is in fact a basis.

The standard topology on the real numbers R is invariably defined via a basis. You define the basis B = {(x-e,x+e): x in R and e > 0}. Then the topology on R is the topology generated by these sets. I.e. a subset U of R is "open" (a region) if and only if it is a union of the above open intervals, if and only if for every x in U there exists e > 0 such that (x-e,x+e) ⊆ U.

The standard topology on R^n (such as n = 2,3,..) is done the same way, but instead you define "spheres" as N(x,e) = {y in R^n: |x-y| < e}, where |x-y| = √((x1-y1)^2 + ... + (xn-yn)^2). Then B is defined as {N(x,e): x in R^n and e > 0}. Then a subset U of R^n is open if and only if for every x in U, there exists e > 0 such that N(x,e) ⊆ U.

In both the cases of R, R^n, you must check that the property (***) in 1.4 is satisfied by B (exercise).

Now after a hefty serving of abstractions, we are now in a position to discuss connectedness...

 

[rudinreader]

Well, anyways, it's not going to be the end of the world to finish what I started. This does at this time sort of feel like a "write your own lecture on general topology" undergraduate assignment...

Ultimately, the punchline is --- at the time Calculus begins you assume that R is a connected space, and then prove the completeness axiom... And the advantage, whether or not it's worth it, is that somehow this can help.

Connectedness

In the (finite) geography example of X = {HI,WA,AK,R}, S = {{HI},{WA},{AK},{WA,AK,R},X}, we are ultimately interested in using topology to distinguish the elements of X as either being a "base" (or in this case a U.S. State) or a "connection" between bases. This "connection" could represent the "ferry from Washingon to Alaska", the "flight routes" of an Airline between cities, or even a border (geometrical abstract line) between states (or even the border between the "geological regions" of land and water). We will need the following concepts.

Definition 1.5: If U is any subset of X, then

1) The union of all the regions that are subsets of U is the interior of U
2) The union of all the regions in the complement X \ U is called the exterior of U
3) The set of all points that are not in the interior or exterior of U is called the boundary of U.

Again, we are often concerned with having a practical way of "identifying" these types of sets in X, based on our construction of the topology. When you apply Theorem 1.3, you can identify these sets as follows (proof as exercise):

1) The interior of U is the union of all the bases in U
2) The exterior of U is the union of all the bases in X \ U
3) x is in the boundary of U if and only if every base containing x intersects U and X \ U

We will denote the interior, exterior, and boundary of U as int(U), ext(U), and bdy(U) respectively.

We consider these concepts as in the geography example. In the first example, we note (exercise) that if U is a region, then int(U) = U. And if the complement X \ U is a region, then ext(U) = int(X\U) = X\U.

int({HI}) = {HI} is a base (hence a region)
ext({HI}) = int(X\{HI}) = X\{HI} = {WA,AK,R} is a base
bdy({HI}) = bdy(X\{HI}) = {} (empty).

int({WA,R}) = {WA}
ext({WA,R}) = int(X\{WA,R}) = X\{WA,R} ={HI,AK}
bdy({WA,R}) = {R}

int({R}) = {}
ext({R}) = int(X\{R}) = X\{R} = {WA,AK,HI}
bdy({R}) = {R}

int(X) = X
ext(X) = int({}) = {}
bdy(X) = {}

As has been hinted in this example (proof as exercise), for any set U, int(U) = ext(X\U), and bdy(U) = bdy(X\U) = X\(int(U)∪ext(U)). In other words, the interior/exterior/boundary concept relate a set U to its complement. We also note (exercise) that int(X) = ext({}) = X, ext(X) = int({}) = {}, and bdy(X) = bdy({}) = {}, for any space X.

Again, our goal has been to mathematically discuss the intuitive concept "connectedness". Aside from the bit of abstraction required in constructing topological spaces in the previous section, the definitions of "interior","exterior", and "boundary" are a bit more straightforward. We can use these simple definitions to describe when a region is connected, or disconnected.

We will say that the region V is a subregion of the region U (possibly X) if V is a nonempty, proper subset of U. In other words, "subregion" is exactly what it would mean in everyday English, we just require that a "subregion" of U is not empty and is not U itself.

Definition 1.6: A region U is said to be connected if it is not disconnected. A region U (possibly X) is said to be disconnected if any of the following equivalent statements is true:

1) Some subregion of U does not have a boundary
2) U is a disjoint union of subregions. I.e. U = V_1 ∪ ... ∪ V_n where V_1,...V_n is a list of disjoint subregions of U.
3) U is a disjoint union of two subregions, i.e. U = V_1 ∪ V_2, where V_1,V_2 are disjoint subregions of U.


For the geography example,

{HI}, {WA}, {AK}, {WA,AK,R} are all connected regions of X (why?).
X = {HI} ∪ {WA,AK,R}, {HI,WA,AK} (X after R is "bombed") = {HA} ∪ {WA} ∪ {AK} are disconnected regions. In particular, X is disconnected so is called a disconnected topological space.

We now wish to enlarge our geography example, so that we can identify particular "connections" between regions. Let HI,WA,AK,R be defined as before. We now wish to add a connection "F" for an airline that has flights connecting HI,WA,AK, and another region "AN" = Antartica, whereas the airline does not fly to Antartica. We define the space and it's bases for a topology as follows:

X = {HI,WA,AK,AN,R,F},
S = {{HI},{WA},{AK},{AN},{WA,AK,R},{HI,WA,AK,F},X}.

In this case, again, {R} and {F} are not bonafide regions, but again will be connections relative to certain subregions of X. We now observe (check these identies as an exercise):

bdy({WA}) = bdy({AK}) = {R,F},
bdy({HI}) = {F},
bdy({AN}) = {}.

It follows that {AN} is disconnected from the rest of X. {HI,WA,AK,F} is a connected bonifide base, and by taking a union, you get "the greater HI-WA-AK area" as {HI,WA,AK,R,F}, also a connected region, but is not a base (because we didn't add it to S). Notice also that with this definition {HI,WA,F},{WA,AK,F},{HI,AK,F} are not regions, because if S yielded this, by taking intersections you would get that {F} itself is a region - which is not what we want! In this way, "F" is like one airplane, you can go everywhere with it, but if it is bombed, you can't go anywhere at all (except the old fashioned R route). If instead you want a space that has ensures flight service in WA-AK even if Japan takes over Hawaii, then you should instead define three flight routes F1,F2,F3, and define the bases as {HI,WA,F1}, {WA,AK,F2}, and {HI,AK,F3}. But since this type of connection is the same as R, we will get a more general taste by considering just one F defined as above.



Overall Summary of the Geography Example

In the finite example we want to define "bases" as points and also "connections" as points of the space X. Then what remains, is that we want to be able to see how the topology characterizes the "regions" vs the "connections" while the set X on it's own does not.

Well, the key observation is that {HI,AK,WA} is not connected, but {HI,AK,WA,F} is, and similarly for {WA,AK}, {WA,AK,R}. If V = V1∪...∪Vn is a set of disjoint regions, and {P} ∪ V is a connected region, then you can call P a connecting point of V. Evidently, {P} is not a region, because we can't have P in V, otherwise {P} ∪ V = V is not connected. So {P} is not in V, so if {P} is a region, V* = {P} ∪ V is a disjoint union of regions, so is not connected.

But one thing is lacking is that the connecting point is not unique to any given region. We have "F" is a connecting point for {HI}∪{WA}∪{AK} = {HI,WA,AK} and also a connecting point for {HI}∪{WA,AK,R} = {HI,WA,AK,R}. But we would intuitively like to signify that F in particularly connects {HI,WA,AK}.

In general, this is a fact of life. For example, take X = {a,b,c,d,r}, and S = {{a},{b},{c},{d},{a,b,r}, {c,d,r}}. Then X = {a,b,r}∪{c,d} is a disconnected space, but it has two connected regions {a,b,r}, {c,d,r}, and in both cases the connecting point is r. So we can't in general think of a connecting point (as defined above) like a "route" - which was the aim of our intuitive geographical example.

However, in the context of the geographical example which is finite, we can signify the "fundamental region" a connecting point connects by taking the intersection of all the regions it connects. So if it serves as "two connections" as in the X = {a,b,c,..} example, this intersection will be empty. If there are only finitely many such regions - the result will always be a region. So in the geography example, the fundamental region that F connects is {HI,WA,AK}∩({HI}∪{WA,AK,R}) = {HI,WA,AK}, and the fundamental (and only) region for R is {WA,AK}. So in this finite geography example, the topology identifies the "connecting points" as "routes", when we define them in the way we did. It should be straightforward to extend this "finite geography" to other (perhaps more complicated) situations, like the space of cities and connecting airline flights.

Going forward we will begin to pursue these concepts in the direction of the (infinite) space R, and hopefully the abstractions would then be helpful for Calculus.. will bla bla bla..