Hot answers tagged

11

You're mixing the actual number with its representations. Yes, in base $n$, the string $10$ represents exactly $n$. But now forego of the decimal representation of $10$, and think about it as "how many digits a healthy human being has on both their hands". This is your $n$, now. Let's for the sake of simplicity call this number "ten". Now we are counting ...


10

This is more like a comment but it's too long, so I'm putting it as an answer instead, please accept my apology. In base $10$, the "symbol" $78152_{10}$ represents the number $78152_{10}=7\cdot 10^4 + 8\cdot 10^3 + 1\cdot 10^2 + 5\cdot 10^1 + 2\cdot 10^0$. In base $n$, the "symbol" $78152_n$ represents the number $78152_n=7\cdot n^4 + 8\cdot n^3 + ...


8

open cover (for a given topology $\tau$): a collection of open sets $U_{\alpha}$ (open with respect to the topology $\tau$) such that $$X \subset \underset{\alpha \in A}{\bigcup} U_{\alpha}$$ open set: A set in the topology $\tau$ of the space $X$. compact (for a given topology $\tau$): every open cover of $X$ (with respect to the topology $\tau$) has a ...


5

You should use elements of the topology (open sets) to cover your space.


5

Also in your lecture notes should be: Let $X$ be a set, $(X,\mathcal{T})$ a topological space. Then a collection $\mathcal{A} \subset \mathcal{P}(X)$ is called an open cover if: (i) $\mathcal{A}$ is a cover, i.e. $\forall x \in X, \exists A \in \mathcal{A}$ such that $x \in A$. (ii) Elements of $\mathcal{A}$ are open, i.e. $\mathcal{A} \subset ...


4

Here, "resp." is an abbreviation for "respectively". So: If $U$ and $V$ are open (resp. closed) then $U\cup V$ is open (resp. $U\cap V$ is closed). means: If $U$ and $V$ are open (respectively closed) then $U\cup V$ is open (respectively $U\cap V$ is closed). which is a lazy way to write: If $U$ and $V$ are open then $U\cup V$ is open. If $U$ ...


4

See Restricted quantifiers: $(\forall x \in D)P(x)$ is equivalent to: $\forall x (x \in D \to P(x))$. Thus, your formula: $∀x \in U : f(x) \in V$ is equivalent to: $\forall x (x \in U \to f(x) \in V)$.


3

a) An algebraic subset $X\subset\mathbb C^n$ has all its irreducible components of dimension $n-1$ if and only if it is the zero set $X=V(P)$ of some polynomial $P\in \mathbb C[T_1,...,T_n]$. b) Beware that $X=\{(1,0)\}\cup V(T_1)\subset \mathbb C^2$ is an algebraic subset of dimension $1$ (=the maximum dimension of its irreducible components) but ...


3

Let's analyse this logically, because what else do we do in mathematics? There exists $\delta$ for all $\epsilon$ This entails that for some $\delta$ we have that it is independed of our $\epsilon$. That means if $\epsilon=1$ or $\epsilon=10^8$ we have the same $\delta$, which is bad because it can be suddenly at any difference. For example ...


2

An open cover is a collection of open sets whose union is $X$. Thus it is a subset of $\tau$ to begin with. Since every subset of $\tau$ is finite, then so is every open cover.


2

When I work with numbers in base 16 or base 2, the string "10" is not called "ten", but "one zero". We generally don't have names for specific integers that reflect some other base, so although a full 16 bit value is ffff and then 1 0000 is significant to me, I don't have a spoken name for it that corresponds to thousand. It's only known as "sixty ...


2

The name refers to the way we choose to group our items. The way we've evolved, we found that nine counting symbols and a symbol for nothing suffice for our preferred base. We're able to recycle the glyphs $1$ and $0$ to denote our grouping, which is given by the combined symbol $10$. To illustrate why it's all about the grouping, and not about the final ...


2

You have already solved your question when you finish the use of triangle inequality. Why do you want to go for other values of $\epsilon$'s? The question is clearly not talking about any $\epsilon$. It is just asking you to find a $\delta > 0$ such that some specific statement containing $\delta$ is true. You have already shown that such a $\delta$ is ...


1

A vector space is a mathematical structure, and while in itself is quite informative, once you know a set is a vector space it doesn't mean your work is done, you can stop burning the midnight oil and hit the hay. For example: there's an obvious vector space isomorphism between the space of $n-1 $ degree polynomials (call $\mathbb{R}[x] $) and ...


1

The formal definition of a vector is pretty open ended (a member of a vector space). At a very high level vector is a collection of mathematical objects, that obeys rules of addition and scalar multiplication. A container of numbers isn't too bad. But the objects could be something like differential operators. And, they could be other vectors. But then, ...


1

Suppose $F$ is homeomorphic to a compact subspace of $\Bbb R^\infty$. (There is some nice classification of which spaces this is true of, but I forget it. Maybe every compact Hausdorff space? At the very least it is true of any compact manifold and of any finite CW complex.) Consider the space "$B(F)$" (terrible notation, sorry; this is meant to evoke ...


1

Consider the difference between $~\forall \epsilon~\exists \delta~( \epsilon < \delta)~$ and $~\exists \delta~\forall \epsilon~(\epsilon <\delta)~$.   Are these equivalent statements? The first statement says: "For every number there is a number that it is less than."   This is true; because there is no maximum real number. The second ...


1

A connected component is usually only defined on undirected graphs. (see below). For the more general digraphs we use the concept of strongly connected components (see below) which generalize the notion of connected components to directed graphs. Sometimes we also talk about weak connectedness which is just the (undirected) connectedness if when we consider ...


1

If you define the degree of a polynomial as the degree of the highest non-zero power, then if the polynomial is zero, the degree is undefined. You could define it by convention, to make sense of some general rules, and then as the other answers explain, you get different results. Now, think of why $x^0 =1$. The proof is by using the laws of exponents, ...


1

Convention 0. Marc has already explained why $\mathrm{deg}(0) = -\infty$ is a good convention. Convention 1. On the other hand, if we want $\mathrm{deg}(PQ) \geq \mathrm{deg}(Q)$, well this implies $\mathrm{deg}(0) \geq \mathrm{deg}(Q)$ for all $Q$, so therefore $\mathrm{deg}(0) = +\infty$ is a good convention. Just take a look at the following divisibility ...



Only top voted, non community-wiki answers of a minimum length are eligible