oh yes, I am supposed to look at the graph of f. I guess I was just mistaking the graph of f for the image of f. I guess I'm not seein gthe distinction between the graph and image.
I know this proof is probably super easy but I'm really stuck. I don't want someone to solve it for me, I just want a hint.
One way is trivial:
suppose f continuous.
[0,1] compact and the continuous image of a compact space is compact so f([0,1]) is compact
Now the other...
I was wondering if someone can tell me if my approach to the proof is a correct one. (rather than typing it all out here and making a mess of the notation, I typed it up in latex and did a screencap then put that on imgur, so the following link has the proof)
http://i.imgur.com/gNFToKx.png
In reading about the Tube Lemma, an example is given where the Tube Lemma fails to apply: namely, the euclidean plane constructed as R X R. The Tube Lemma does not apply here because R is not compact. The example given is as follows:
Consider R × R in the product topology, that is the...
I am new to group theory, and read about a "universal property of abelianization" as follows: let G be a group and let's denote the abelianization of G as Gab (note, recall the abelianization of G is the quotient G/[G,G] where [G,G] denotes the commutator subgroup). Now, suppose we have a...
oooooh yeah. I'd forgotten about the product of covering spaces being a covering space for the product. Well, at least it makes sense how to understand think of it in one direction. But I'm still confused about something: am I wrong in thinking that the covering spaces of the torus can be...
Thanks bacle. The good thing for me is that I am probably more comfortable with the concept of induced homomorphisms then anything else I've come across so far in the last bit of algebraic topology. Thanks a lot for the third paragraph - I feel like this is kind of the logic I'd been using...
Hey guys, thanks for the responses. The reason I am confused is because, the torus T^2 has fundamental group isomorphic to Z X Z. Now, it's a result that if a space X' covers the torus, then X' is homeomorphic to either R^2, S1 X R, or T^2. But I know that the isomorphisms of covering spaces...
EDIT:( Sorry I meant for the title of this to say conjugacy classes of subgroups of Z X Z) I have a question.I am trying to figure out, what are all of the normal subgroups of Z X Z? Well, I know that Z has the following subgroups: the trivial group {0}, itself Z, and then aZ, one for each a in...
I am just learning about covering spaces and I feel almost every theorem i have to work with starts something like "if you have a covering space p:(\tilde{X}) -> X..." etc. I am a little lost because I'm wondering how I look at a space and then say to myself, what are the possible spaces that...
I am just trying to figure out how to make a CW complex for this. For the n-genus orientable manifold (connect sum of n-tori) I feel like a lot of things make sense, fundamental group, CW complex, etc. But in the infinite case, things seem to fall apart. For example, I can not figure out how...
Boundary of a Mobius band - I think S1 V S1, everyone else says S1??
Hey I am having a huge problem! There are a few problems where I'm using Van Kampen's theorem and for one part of the problem I need to compute the fundamental group of the boundary of the Mobius band. Everyone keeps telling...