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Doing math.


Dec
6
comment Proof for Image of Indexed Collection of Sets?
@Randomstop. Ideally you would want that $y\in f(\bigcap_{\alpha\in\Lambda} U_{\alpha})$. But you already know that $y=f(x)$ where $x$ is a member of all $U_{\alpha}$ (by considering $x=x_{\alpha}=x_{\beta}=..$). So you're practically almost there.
Dec
6
comment Proof for Image of Indexed Collection of Sets?
@Randomstop. Yep, you got it.
Dec
6
comment Proof for Image of Indexed Collection of Sets?
@Randomstop. Exactly. So in this scenario you know that $f(x_{\alpha})=y$ for all $\alpha$. So, if you take any indices $\alpha,\beta$, what do you know of $f(x_{\alpha})$ and $f(x_{\beta})$? And what does that tell you about the relationship of $x_{\alpha}$ and $x_{\beta}$?
Dec
6
comment Prove that $\lim_{(x,y)\to(0,0)} \frac{x^3y}{x^6+y^2} = 0$
Is $\frac{x^{3}y}{x^{6}+y^{2}}$ always non-negative? Even for points arbitrarily close to the origin?
Dec
6
comment Proof for Image of Indexed Collection of Sets?
@Randomstop: Since $x_{\alpha}$ is in the domain and $y$ in the range, we can't necessarily say that $x_{\alpha}=y$. But does the injectivity of $f$ imply something about the relationship of each $x_{\alpha}$ to each other?
Dec
6
revised Cardinalities of power sets, $\mathbb{N}$, and $\mathbb{R}$.
tex edits
Dec
6
revised for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
tex edits
Dec
6
revised Finding a continuous function $f:(X,T_u)\to (X,T_e)$ with $f(0)=0$ and $f(1)=1$ that has some $c\in (0,1)$ such that $c\notin f(X)$.
some tex edits
Dec
6
answered Proof for Image of Indexed Collection of Sets?
Sep
24
comment How to apply Borel-Cantelli Lemma here?
I edited it to have homework tag. Correct me if I'm wrong.
Sep
24
revised How to apply Borel-Cantelli Lemma here?
edited tags
Aug
28
comment A set $A$ is closed iff $\operatorname{fr}A\subseteq A$.
@Citizen. What definition of an exterior point do you use then?
Aug
28
comment A set $A$ is closed iff $\operatorname{fr}A\subseteq A$.
is $Fr(A)$ the closure of $A$?
Aug
28
answered How can I prove that this set is finite?
Aug
23
revised When is the logarithm of this function square integrable?
edited title
May
19
awarded  Constituent
May
7
awarded  Caucus
Mar
21
comment “Maximal” sets with cardinality $\aleph_0$?
Few comments and questions: 1. What do you mean by being dense in itself? Dense in its own subspace topology? In this sense every set is dense in itself. 2. Denseness is a topological property. Saying that $\mathbb{Q}$ is more dense or less dense doesn't make sense: it either is dense (in some superset) or it isn't.
Mar
20
comment Set of Continuous Functions, Functionals, and Equicontinuity
What are your thoughts on this problem? Have you tried anything?
Mar
14
comment I thought I proved the Limit here didn't exist…
Note that $\sqrt{2x^{2}}=|x|\sqrt{2}$ and not $x\sqrt{2}$.