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answered If $K\in \Bbb{R}^n$ is compact, $\sup_{x,y\in K}|x-y|=\max_{x,y\in K}|x-y|$.
Jun
27
comment How is the general concept of a basis useful?[topology]
Finally, one of my favourites: show that a space $Y$ is a Hausdorff space if and only if it has the closed graph property: for any space $X$ and any continuous map $f\colon X\to Y$, the graph $\Gamma_f=\{(x,f(x))\colon x\in X\}$ is a closed subset of $X\times Y$.
Jun
27
comment How is the general concept of a basis useful?[topology]
For some exercises, prove that the product topology on $\mathbb R^n$ is the same as the (Euclidean) metric topology. Show that the product of two Hausdorff spaces is a Hausdorff space and that the product of two compact spaces is compact. Suppose $X,Y$ are topological spaces, and that $A\subset X,B\subset Y$. Show that the subspace topology on $A\times B$, considered as a subspace of the product $X\subset Y$, is the same as the product topology on $A\times B$, where $A$ is considered as a subset of $X$ and $B$ as a subset of $Y$.
Jun
27
comment How is the general concept of a basis useful?[topology]
It's hard to answer your question without knowing what level you're at. In any case, you'll get a much better idea about bases of topological spaces by doing some actual topology than you will from any answer posted to your question. To get a good understanding of bases of open sets, study metric spaces (a basis of open sets for a metric space is given by the collection of open balls in that space) and product spaces (where the topology is usually defined by giving an appropriate basis).
Jun
26
answered Is the complement of a closed set always open?
Jun
24
comment Showing a set of box topologies is homeomorphic
Just as a comment - it's perfectly acceptable to write $f(x,y)=(y,x)$. If you like, you can then observe that $y=\pi_Y(x,y)$ and that $x=\pi_X(x, y)$, so it's clear that the maps are continuous. Otherwise, what you've written is completely correct.
Jun
24
reviewed Reject Black and white setting mathematics lectures
Jun
23
comment If a point has no dimension and no area how can there be space?
The length of a line is certainly not equal to the cardinality of the set of points making it up. Why on earth do you think that?
Jun
23
answered Question regarding an unsolved problem involving a trigonometric sequence
Jun
23
revised Can $n^2+4n$ be a perfect square?
added 1 character in body
Jun
22
answered Can $n^2+4n$ be a perfect square?
Jun
22
answered When the logarithm and integral can be commuted?
Jun
22
comment Prove that there exists a linear transformation $T: R^n \rightarrow R^n$ such that $T^3 = T$, and T has at least three distinct eigenvalues.
Your instincts are right. We might as well take $C$ to be the identity matrix and let $B$ be $3\times3$ satisfying the conditions in the question. This means we can reduce immediately to the $3\times3$ case.
Jun
22
comment Prove that there exists a linear transformation $T: R^n \rightarrow R^n$ such that $T^3 = T$, and T has at least three distinct eigenvalues.
@Chilango That's not what I had in mind. I'm aware that the exercise is to produce examples; that's why this is labelled as a hint, rather than a complete answer.
Jun
22
revised Prove that there exists a linear transformation $T: R^n \rightarrow R^n$ such that $T^3 = T$, and T has at least three distinct eigenvalues.
deleted 58 characters in body
Jun
22
answered Prove that there exists a linear transformation $T: R^n \rightarrow R^n$ such that $T^3 = T$, and T has at least three distinct eigenvalues.
Jun
21
comment Taking it a step further with a sum
@StevenTaschuk Feel free to edit it so it's correct.
Jun
21
revised Taking it a step further with a sum
Fixed it up
Jun
21
comment Show that there exists a coordinate system in $\mathbb{P}^n$ such that $P_0=(1:0:\cdots:0),\ldots,P_n=(0:0:\cdots:1),P_{n+1}=(1:\cdots:1)$
1. $u_0,\dots,u_n$ is a basis for $\mathbb K^n$, and $u_{n+1}$ is a non-zero vector (by definition of homogeneous coordinates in $\mathbb P^n$).
Jun
20
revised Taking it a step further with a sum
deleted 16 characters in body