Janson A.J
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Next privilege 250 Rep.
 Sep24 awarded Autobiographer Sep13 answered $f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$ Aug21 accepted How to define $a^x$? Aug21 asked How to define $a^x$? Aug20 comment Is $\{x\}$ a neighbourhood of $x$? That depend on our topology. I think you're studying metric spaces only. If you don't know general topology it doesn't matter. Actually by a topology we mean we first set the open sets. I mean we just define that these are the open sets and do maths on that. So you can define a topology on a set in which one particular singleton set is open another one is not open, like that.. So {a} is open in your topology iff {a} is a neighborhood of a. As other people have already mentioned, there is a topology called discrete topo. in which all singleton sets are open. => {x} is a nbd of x for all x..!:) Aug20 comment Is $\{x\}$ a neighbourhood of $x$? Any open set containing our point 'x' is a neighborhood of x. Don't confuse with the real life meaning of 'neighborhoods'. It has no connection with saying that points in a neighborhood of x are closer to x or things like that. Even the whole space itself is a neighborhood of any point! Aug20 answered Sphere-sphere intersection is not a surface Aug20 answered Show that following subset of $\mathbb R^2$ is compact Aug17 answered Prove that invertible metrices set is an open set in a given space, and the determinant is continuous Aug17 answered Basic compactness Aug17 awarded Teacher Aug17 answered How does one represent a range like $[a,b]$ if the ^range^ is exactly $1$? Aug16 awarded Editor Aug16 revised When discussing compactness, is it necessary to specify the metric space? added 335 characters in body Aug16 answered When discussing compactness, is it necessary to specify the metric space? Mar17 awarded Supporter Mar17 asked Can we generalize the result of Urysohn's lemma to countable collection of pairwise disjoint closed subsets of a normal space..? Mar17 awarded Scholar Mar17 accepted Convergence of the series $\sum_{n=1}^{\infty}((1/n)-\sin(1/n))$..? Feb26 awarded Student