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May
13
comment Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then 1. $Cl(A) = Cl(Int(A))$ 2. $Int(A) = Int(Cl(A))$
@AlyssaWallace: No problem. I'm glad you found it helpful.
May
13
comment Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then 1. $Cl(A) = Cl(Int(A))$ 2. $Int(A) = Int(Cl(A))$
@AlyssaWallace. Sure. You're welcome
May
13
comment Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then 1. $Cl(A) = Cl(Int(A))$ 2. $Int(A) = Int(Cl(A))$
@AlyssaWallace: The whole real line is an open set so its interior is the whole real line itself.
May
12
comment boundary of $\mathbb{Q}^n$ in $\mathbb{R}^n$
$\partial\mathbb{Q}$ is not $\mathbb{Q}$.
May
12
comment Prove that $X$ contains exactly two clopen sets if and only if every nonempty proper subset of $X$ has a nonempty boundary.
This looks correct. In the first direction you probably don't mean that it's closure is empty but boundary. Maybe a typo.
May
12
comment Sum of open/closed/compact sets in $\mathbb{R}^n$ open/closed/compact
@JackLee: Right, good call. I totally overlooked that one. Thanks for pointing it out, I'll edit it.
May
11
comment injective/path component
@smith: So you have concluded that $\pi_0(\{0,1\})$ is a $2$-point set, and $\pi_0([0,1])$ is a $1$-point set. Now, there is only $1$ map from a $2$-point set to a $1$-point set. What is it? Is it injective?
May
11
comment injective/path component
@smith: Once you know what $\pi_0(\{0,1\})$ and $\pi_0([0,1])$ are in this particular case, you will realize that there will only exist one map from the first to the latter. This must precisely be the induced map: you don't even need to calculate it explicitly.
May
11
comment injective/path component
@smith: Exactly. So what are $\pi_0(\{0,1\})$ and $\pi_0([0,1])$? Now note that any map defined on a discrete set is continuous. The only injective ones is the identity map and the permutation map. Take the identity map for example. What must the induced map $f_*$ be? And why?
May
11
comment injective/path component
@smith: What are the clopen sets of the discrete $2$-point set? What about the interval $[0,1]$?
May
11
comment injective/path component
@smith: The unit interval is path connected within the reals.
May
10
comment Path components
@smith: Yes, it won't be connected. But the two components are path components. Note that $f_+$ and $f_-$ are continuous functions and the sets are continuous images of path connected sets.
May
10
comment Example 3, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Why is the topologist's sine curve not locally connected?
@SaaqibMahmuud: For each $k$, we have $\emptyset\neq A_k\neq I_k$, so $I$ is just a countable union of disjoint disconnected sets.
May
7
comment what are these Analytic functions?
What do you mean by a real time example?
May
7
comment Why is the set of integers in $\mathbb{R}$ closed?
@orangeskid: Discrete as having the discrete topology. I.e. every singleton is an open set. The metric that gives this topology, for instance the so called discrete metric, is defined by $d(i,j)=1-\delta_{i,j}$.
Mar
29
comment Let $X$ be a topological space. Prove that for any $x $ in the intersection of all opens sets $=\{x \}$, the space $X$ need not be Hausdorff.
This is answering a different question than what you formulated in the problem statement in the comments. Saying that the intersection of all open sets around a point will yield the singleton, is not the same as saying that any two open sets containing a point will yield the singleton. With Zariski topology the intersection of two open sets is always uncountably infinite, while with uncountably infinite intersections you can get the singleton.
Mar
27
comment A problem in Weierstrass M-test proof
For a sequence $(a_{n})_{n=1}^{\infty}$, if $|\sum_{k=1}^{N}a_{k}|\leq \sum_{k=1}^{N}|a_{k}|$ for all $N\in\mathbb{N}$ by the triangle inequality, then taking limits on both sides you get $|\sum_{k=1}^{\infty}a_{k}|\leq \sum_{k=1}^{\infty}|a_{k}|$, if both limits exist. So what is the problem of using triangle inequality infinitely many times?
Mar
27
comment finding formula for M^n
What exactly is the matrix $M$ here? Are those the column or the row vectors of $M$ that you have given there?
Mar
27
comment Proof by contradiction to prove an inequality does not hold
If you want to show that there is no positive integer such that ... , the counter assumption is that there is such a positive integer. Not, that it is true for all positive integers. As for a hint how to proceed: as Andre Nicolas said you can just divide by $x$ to get a contradiction.
Mar
27
comment Define Radon measure as an integral
What is the definition of an outer Radon measure that you're working with?