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Jan
18
answered How do i prove that this given set is open?
Jan
18
asked How do i prove that this given set is open?
Jan
17
comment What are 'weak' forms of Urysohn's lemma, which do not require choice?
However, I'm not sure Urysohn's lemma holds for locally compact space. Is it provable?
Jan
17
comment What are 'weak' forms of Urysohn's lemma, which do not require choice?
@gnometorule I just spent my whole day to read the argument and whole references in the link related to Urysohn's Lemma, and it works fine without any choice! And more surprisingly, "If $X$ is a regular Hausdorff and second countable and $A,B$ are disjoint closed subsets, there exists a 'uniquely' defined continuous function $f:X\rightarrow [0,1]$ such that $f(A)\subset\{0\}$ and $f(B)\subset\{1\}$" is true in ZF
Jan
17
comment What are 'weak' forms of Urysohn's lemma, which do not require choice?
@Martin Yes, exactly. How do i call $d(x,A)$ then?
Jan
17
comment What are 'weak' forms of Urysohn's lemma, which do not require choice?
@gnometorule Are you even sure for the argument in the link? Otherwise i'm going to spend my whole day to check it.
Jan
16
revised What are 'weak' forms of Urysohn's lemma, which do not require choice?
added 76 characters in body
Jan
16
asked What are 'weak' forms of Urysohn's lemma, which do not require choice?
Jan
16
accepted How do i prove that $C_c(X)$ is a vector space?
Jan
16
asked How do i prove that $C_c(X)$ is a vector space?
Jan
16
comment What is your definition for neighborhood in topology?
Would you please answer my comment above?
Jan
16
comment What is your definition for neighborhood in topology?
@Zhen I'm completely unfamiliar with the definition in wikipedia. Does the existence of a neighborhood (wikipedia definition) of $x$ gurantees the the existence of a open set containing $x$ then?
Jan
16
accepted What is your definition for neighborhood in topology?
Jan
16
asked What is your definition for neighborhood in topology?
Jan
15
comment What is wrong in my proof? (Uniform convergence and Lebesgue integral)
I want to make it clear. Is "$f$ is Lebesgue Integrable" same as saying "$f\in L^1(\mu)$"?
Jan
15
accepted What is wrong in my proof? (Uniform convergence and Lebesgue integral)
Jan
15
comment What is wrong in my proof? (Uniform convergence and Lebesgue integral)
@Ilya Now i got it thank you!
Jan
15
comment What is wrong in my proof? (Uniform convergence and Lebesgue integral)
@Ilya Oh right.. So the fake part is the last part. That is, [$|\int_X f_n-f d\mu|=0 \text{ implies } \int_X f_n d\mu - \int_X f d\mu =0$] is false, since it might me $\infty-\infty$. Am I right?
Jan
15
asked What is wrong in my proof? (Uniform convergence and Lebesgue integral)
Jan
15
accepted Is there an abbreviation for “almost all $x\in X$”?