- #1
rudinreader
- 167
- 0
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!
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...
Last edited: