Do we really need basis? More specifically, there are different notions of basis in mathematics, and the tag is practically useless without adding a distinction between them.

I want to prove that the Order topology on $\mathbb{R}$ has the same basis as as the Euclidean topology on $\mathbb{R}$. Assume that the only thing we know about the order topology is that it has the open rays as its subbase. My problem is that at some point I have to make some kind of claim tha...

Consider our manifold to be $\mathbb{R}^n$ with the Euclidean metric. In several texts that I've been reading, $\{\partial/\partial x_i\}$ evaluated at $p\in U \subset \mathbb{R}^n$ is given as the basis set for the tangent space at p so that any $v\in T_pM$ can be written is terms of them. The ...

Using basic open sets of $\Bbb R$, prove that $f(x,y,z)=x^2+y^2+z^2+2x+2y+6$ is a continuous function from $\Bbb R^3$ to $\Bbb R$. My attempt: Since $f(x,y,z)$ is continuous and $f(x,y,z)\in \mathcal B$ , where $\mathcal B $ is the basis of $\Bbb R$, (I.e. the collection of all open intervals $...

Show that the collection of all open intervals $\{(a,b)\}$ is a basis of $\Bbb R$ with the standard topology: My attempt: I believe we want to show two things: 1) All elements, $x\in\Bbb R$ are contained in some basis element: $\forall x\in\Bbb R$ $x\in(x-1,x+1)$ $\square$ 2) If $x\in B_1\...

In the paper 'Self-taught learning: transfer learning from unlabeled data' by Raina et al, the authors define a term as follows: 'aj(i) is the activation of basis bj for input xu(i) I have not encountered the term 'activation of a basis' before, and have been unable to find a definition of ...

We know that there are $3$ types of $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$. $\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le C_qA^kk^{k\alpha}\qquad (k,q=0,1,2,...)$ $\mathcal{S}^\beta: |x^k\varphi^{(q)}(x)|\le C_kB^qq^{q\beta}\q...

Is there any expression for the integral of the product of three Meijer-G-functions, where the domain of integration is [0;1]?