Janson A.J
Reputation
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 Nov 17 comment graphs of functions which are closed, but fail to be continuous Ohh.. We just have to take the same function on $[-1,1]$.. Nov 17 comment graphs of functions which are closed, but fail to be continuous What if we want $X$ to be compact? ie, I need a discontinuous function $f:X \longrightarrow \mathbb{R}$, whose graph is closed. May 26 answered The sum of an uncountable number of positive numbers Sep 24 awarded Autobiographer Sep 13 answered $f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$ Aug 21 accepted How to define $a^x$? Aug 21 asked How to define $a^x$? Aug 20 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..!:) Aug 20 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! Aug 20 answered Sphere-sphere intersection is not a surface Aug 20 answered Show that following subset of $\mathbb R^2$ is compact Aug 17 answered Prove that invertible metrices set is an open set in a given space, and the determinant is continuous Aug 17 answered Basic compactness Aug 17 awarded Teacher Aug 17 answered How does one represent a range like $[a,b]$ if the ^range^ is exactly $1$? Aug 16 awarded Editor Aug 16 revised When discussing compactness, is it necessary to specify the metric space? added 335 characters in body Aug 16 answered When discussing compactness, is it necessary to specify the metric space? Mar 17 awarded Supporter Mar 17 asked Can we generalize the result of Urysohn's lemma to countable collection of pairwise disjoint closed subsets of a normal space..?