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May
18
reviewed Approve Probability of event in join sample space of X & Y
May
16
comment Understanding the generator of $H_1 (S^1 \times I)$
Also, this question is closely related to the very first question I ever asked on this site: math.stackexchange.com/questions/50362/…
May
16
comment Understanding the generator of $H_1 (S^1 \times I)$
This kind of continuous map that is not invertible is really important to understand homology. For example, how do we know that a loop $k$ that goes twice around $S^1\times 0$ is homologous to $2h$? Because $2h-k$ is the boundary of a map from a single triangle into $X$ that squishes the whole triangle into $S^1\times 0$.
May
16
comment Understanding the generator of $H_1 (S^1 \times I)$
Yes, exactly. E.g. the map from the square to the cylinder obtained by gluing a pair of opposite edges together is a continuous map, but it doesn't have an inverse.
May
16
answered Understanding the generator of $H_1 (S^1 \times I)$
May
16
comment Understanding the generator of $H_1 (S^1 \times I)$
+1 I think this is a beautiful question.
May
15
revised If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?
added engagement with Jake Levinson's comment
May
15
comment If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?
@JakeLevinson - you're right! $K$ is superfluous! I thought through the proof of the snake lemma before applying it but I mistook the fact that the first term in the top row is used at all with the need for it to be exact there.
May
14
revised If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?
clarification
May
14
asked If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?
May
6
awarded  Nice Question
May
4
comment Fractal dimension of a dense subset
I buy this in essence, although there's something to be careful of since I am defining the dimension in terms of $\limsup$ and not assuming that a limit exists...
May
4
accepted Fractal dimension of a dense subset
May
2
comment $R$ commutative ring with unity. Prove if $R/M$ is a field $\implies M$ is a max ideal.
You mean $I\triangleleft R$. Use \triangleleft.
May
2
comment Fractal dimension of a dense subset
@Tryss - Ah! I think you're suggesting I note that open $\varepsilon$-balls contain closed $\varepsilon/2$-balls, etc.
May
2
comment Fractal dimension of a dense subset
@Tryss - that's more or less exactly the question, isn't it?
May
2
comment Does every cover have an irredundant subcover?
But the Zorn's lemma approach still goes through! What point-finiteness does is to force the poset of subcovers to be closed under taking intersections of chains, thus ascending chains have upper bounds. If $\mathcal{F}_1\prec\mathcal{F}_2\prec\dots$ is an ascending chain of subcovers (i.e. descending under inclusion), and the intersection is not a cover, there is an uncovered $x\in X$. Then the set of $k$ for which some set of $\mathcal{F}_K\setminus\mathcal{F}_{k+1}$ contains $x$ must be cofinal in $\mathbb{N}$. But this implies infinitely many sets of the cover contain $x$.
May
2
comment Does every cover have an irredundant subcover?
Actually I think I'm waving my hands my aside here. Let $X=\mathbb{N}$ with the discrete topology, let $\Lambda = \mathbb{N}\times \{0,1\}$, and let $U_{(n,i)} = \{n\}$ for both $i=0,1$. Let $\mathcal{F}_k = \{U_{n,1}\}_{n\geq k} \cup \{U_{n,0}\}_{n\geq 1}$. Then $\mathcal{F}_1\prec \mathcal{F}_2\prec\dots$ is a nonterminating ascending chain in the poset of subcovers.
May
2
asked Fractal dimension of a dense subset
Apr
29
awarded  Nice Question