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2h
reviewed Edit Proof verification: Munkres Theorem 22.1
2h
revised Proof verification: Munkres Theorem 22.1
Little fix.
7h
answered How to prove any closed set in $\mathbb{R}$ is $G_\delta$
7h
comment How to show that countable union of $F_\sigma$ is $F_\sigma$
The $U_{i,j}$ should be closed, not open.
10h
comment How $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln 2$?
Use Euler's criterion (sequence is decreasing in absolute value and alternating) to see that the series $\sum_n \frac{(-1)^n}{n}$ converges. Then Abel's theorem says that this is indeed the value for $x=1$.
10h
revised How $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln 2$?
deleted 21 characters in body
11h
comment I like to know if there is approximated expression of something
The right hand side is "almost" $e^x$, so this expression should be close to $1 = e^{-x}e^{x}$..
11h
answered Find a unique value for $d$ in $(d \cdot e) \pmod{F} \equiv 1$
13h
comment Is it possible to construct Hausdorff compact topology on every set?
Note that $Y$ is discrete and $X$ is its one-point compactification (if $Y$ is infinite).
1d
revised Showing that $Im(L^*)=(Ker\: L)^\perp \space \:\mathrm and \:\:Ker(L^*)=(Im\: L)^\perp$
edited tags
1d
comment True or False: there is a space $X$ such that $S^1$ is homeomorphic to $X\times X$
@Szmagpie No, it's not that hard, I can write a proof as an alternative answer tomorrow, time permitting. It uses basic connectedness arguments. But the algebraic topology approach is also instructive. Both show ideas that are more widely applicable.
1d
revised Relation betwen the base of a product topology and the bases of the topological spaces
added 1 character in body
1d
answered Relation betwen the base of a product topology and the bases of the topological spaces
1d
comment Is the space of subsets of $\mathbb R^n$ with the Hausdorff metric separable?
The topology the Hausdorff metric generates is actually also known as the Vietoris topology, and for this it's also quite easy to see the separability. But this comes at a cost: prove that the metric topology and it coincide. The benefit is that in the Vietoris topology is often easier to some properties like connectedness etc.
1d
comment Property of compact metric space
Note that we only need that $A$ is compact, not that $X$ is.
1d
revised Property of compact metric space
2 typos
2d
comment If $d_1$, $d_2$ are metrics on $X$ find a relationship between $\tau_1$ and $\tau_2$.
E.g. $d_1$ could be the discrete metric, and $d_2$ the usual metric on the reals.
2d
answered Accumulation point implies strict limit point?
2d
revised Accumulation point implies strict limit point?
Latex plus some spelling
2d
comment Help finding paper from the 1920's
You'd have to go to a library, if the papers have not been scanned. Some journals have started doing that, scanning old issues and putting them online. Rend. Del. Circa. Math. di Palermo probably has not. I'm not close to one...