In what sense do you want to look at topology from a categorical point of view? Once you've learned a good chunk of topology you might get something out of the examples in MacLane's categories book, but I don't know of any large-scale categorical exposition of topology (nor do I know what could be gained by such a thing).
I didn't, I learned most of my elementary topology from a class. The book associated to that class was Kinsey, Topology of Surfaces, but we didn't use it much at all.
@Committingtoachallenge Do you remember the movie Wanted (yes, the one with Morgan Freeman, Angelina Jolie, assassins and bending bullets). Remember how the assassins heal themselves? Try that out.
Nah, I'm just kidding. Most of the time the downvotes I get on my answers are from people who hated the question so much they serial downvote everything on the page.
Lots of Downvotes to me are an indication that my answer is wrong. Thankfully, there will always be the kind community member who'd tell me where I went wrong.
Downvotes to my question indicate that I haven't done enough reasearch or my question is too vague or boring. I've written horrible questions (with 10 upvotes and no answers) because sometimes people like the topic more than the question and hence they upvote the topic. Similarly, downvotes can be due to a person disliking a question. So, yes, downvotes do make me feel bad sometimes especially when I've put a lot of effort into a Q/A
@Bobson There is no real official name. I've heard it be called "Krypto" a few times but you can name it whatever you want.
You don't really need to learn vector spaces in a general context if you're just starting out with vectors in the plane or in 3 dimensions, although the properties of a vector space hold there.
well i think is get vectors, and i dont really know the vector spaces look like. but with the subspaces i have made two vector spaces which are $y=x$ and $y=2x$ which should be fine as subspacs
and aparently i can add them together but i don't know what i will get
So it depends on what you mean by "adding vector spaces" @beginner. I'm assuming by that, you mean the subspace $A + B = \{a + b : a \in A \text{ and } b \in B\}$.
Where $A$ is the vector space $y = x$ and $B$ is the vector space $y = 2x$.
I don't define my one-dimensional vector spaces like you did though. I.e. instead of $y = x$, I think about it as the so-called "span" of the vector $(1, 1)$
There is definitely a certain degree of mathematical maturity that you'll need to develop as you advance. It's not something you get in grade school at all, especially if you're in the USA.
I have two subspaces:
$$W_1 = \{(x, 3x) : x\in \Bbb R \}$$ and
$$W_2 = \{(2x, 0): x\in \Bbb R \}$$
How do I get $W_1 + W_2$?
I tried simply adding a sample vector from each, i.e. $$ (1, 3) + (2, 0) = (3, 3)$$ but I don't think this makes sense since this new vector doesn't fit it $W_1$ nor ...
@KajHansen That sounds just like the math here. I think it's better in the US than here, lol. I think I am living in one of the worst places on earth, lol.
I pretend that's an excuse, but I'm actually up this late a ton. My mind functions poorly in the morning and afternoon and very, very well post-midnight. I always like pointing out that the majority of my undergraduate work has been done between 12 AM and 5 AM.
I have two subspaces:
$$W_1 = \{(x, 3x) : x\in \Bbb R \}$$ and
$$W_2 = \{(2x, 0): x\in \Bbb R \}$$
How do I get $W_1 + W_2$?
I tried simply adding a sample vector from each, i.e. $$ (1, 3) + (2, 0) = (3, 3)$$ but I don't think this makes sense since this new vector doesn't fit it $W_1$ nor ...
