#munkres

[more-mathematics-mor-lam]

Stop asking, it’s never happening

[shit-math-teachers-say]

Legitimate mathematical terms: stack of pancakes

[locally-compact-blog]

Every metric is continuous

This fact struck me as very not obvious and super interesting. I thought I’d provide a proof of it here; this is problem 3a of section 20 in James R. Munkres’ Topology, 2nd. Edition.

Proof: Let $X$ be any metric space with some metric $d:X\times X\rightarrow\mathbb R$, and let $U$ be open in $\mathbb R$. Consider any point $(x,y)\in d^{-1}(U)$, and select a basis element $(\alpha, \beta)$ of $\mathbb R$ for which $d(x,y)\in (\alpha, \beta)\subset U$. Pick a number $\epsilon > 0$ such that  $d(x,y)\in (d(x,y)-\epsilon, d(x,y)+\epsilon)\subset (\alpha, \beta)$ (convince yourself that that’s reasonable if you have to--it seems to come in handy in a lot of these kinds of arguments). Now, consider the following sets: $$A=B_d(x, \frac{\epsilon}{2})$$ and $$B= B_d(y, \frac{\epsilon}{2}).$$ Since both $A$ and $B$ are open in $X$ due to the fact that $x$ and $y$ both came from $X$, $A\times B$ belong to a basis for $X\times X$; further, observe that $(x,y)$ itself belongs to $A\times B$. Let $(x_0, y_0)\in A\times B$. On the one hand, we have:

$$d(x,x_0)+d(y,y_0) < \epsilon$$ which implies

$$d(x_0,x)+d(x,y)+d(y,y_0) < d(x,y)+\epsilon$$

and further, by the triangle inequality and symmetry of $d$, that

$$d(x_0,y_0) < d(x,y)+\epsilon < \beta.$$

On the other hand, we have: $$d(x_0, y_0)+\epsilon > d(x_0, y_0)+d(x,x_0)+d(y,y_0)$$ and so

$$ d(x_0, y_0)+\epsilon > d(x,y)$$ and finally

$$d(x_0,y_0) > d(x,y)-\epsilon > \alpha.$$

Therefore $d(x_0, y_0)\in (\alpha, \beta)\subset U$, which means that $(x_0, y_0)\in d^{-1}(U)$, and so $(x,y)\in A\times B\subset d^{-1}(U)$ meaning that $d$ is continuous.                                                                                           ∎

[tunnels-end]

Students often remember this theorem as stating that the collection of compact sets in a metric space equals the collection of closed and bounded sets. This statement is clearly ridiculous as it stands, because the question as to which sets are bounded depends for its answer on the metric, whereas which sets are compact depends only on the topology of the space.

Munkres throws shade at his students

[tunnels-end]

Munkres, you took 177 pages to prove the uncountability of R.

Shame.

[ernestyalumni]

Topology: Connected components, etc.

I'm going to have to look up Munkres' Topology again for explicit and thoroughly elaborated proofs and explanations on connected spaces, connected components, and I'll have to take a look again at it to distinguish and clarify what the differences are between connected components, locally connected, path-connectedness, locally path-connected, arcwise-connected, and locally arcwise-connected.  Whew.

Bredon's Geometry and Topology is a pain here as it is terse on explanations.

Note that these comments are all coming from a physicist.

[fuckyeahtopology]

update

I got through the readings for class Thursday, and the two books could not be more different. Munkres is unbelievably technical, building up set theory in a very rigorous fashion which is cool and all, but a bit hard to see the motivation for at the moment. most of the results are proven, often so far as to make it seem like the rest of the math I've done wasn't proven. e.g. we start with the Axiom of Choice and use that to justify using induction to prove anything. oh, and the principle of induction is no longer that if something is true for 1, and if if it being true for k-1 it is true for k, then it is true for all n, but that any inductive set A of positive integers is equal to Z+, the set of all positive integers. dense, but rewarding. Armstrong, on the other hand, starts straight into the "cool part", with Euler's theorem for polyhedra, namely that for a particular type of polyhedron, the number of vertices minus the number of edges plus the number of faces equals two. this is true for LOTS of polyhedra and is WAY COOL. from there it works up all the way through the classification theorem, which states that ALL compact connected surfaces are homeomorphic to either a) the sphere, b) the sphere with a finite number of handles, or b) the sphere with a finite number of disks replaced by möbius strips. and those are ALL distinct from each other. the only thing is that most of those results aren't even close to proved. but you can't have it all, I guess. class in two days! and then I get to explain to a bunch of people who are smarter than me what surfaces and abstract topological spaces are. wooooo.

[fuckyeahtopology]

This is what I've been immersed in today. Thus far it's been review of set theory, but much more rigorous than any treatment of similar material I've ever seen. This distrust of anything that hasn't been explicitly proven is something that I've come to really like about higher maths, though, so that's nice.

I have a meeting with my prof tomorrow morning to talk about things that are confusing, so I guess I should keep reading and start the problems, as so far the material isn't really that confusing. I think the other text has more new things though, so that may well be the source of the majority of the confusion for me. Who knows.