All questions with tag [math: general-topology]


Dealing with Tychonoff's Theorem.

Here are my few questions that I encountered while going through Tychonoff's theorem in . a) First of all, so far I was thinking that Heine Borel definition of compactness implies sequential compactness but not the other way around ( although i am failing to find some examples to appreciate it). But what wikipedia says is that " but NEITHER...
2022-07-25 17:47:13

Problem connecting Topology and Algebra via Analysis

Let $C(X):=$ Set of all complex/real valued continuous functions. If $X$ is compact then all the maximal ideals in the ring $C(X)$ is of the form $M_{x}=\{f\in C(X): f(x)=0\}$ for some $x\in X$. Is it true that: If all the maximal ideals in the ring $C(X)$ is of the form $M_{x}=\{f\in C(X): f(x)=0\}$ for some $x\in X$ then $X$ is compact.
2022-07-25 17:47:10

Axiom of choice and compactness.

I was addressing an inquiry lately that managed compactness as a whole topological rooms, and also just how compactness falls short to be equal with consecutive compactness unlike in metric spaces. The only counter - instances that took place in my mind called for hefty use axiom of choice: well - getting and also Tychonoff is theory. Can a pe...
2022-07-25 17:47:06

If the closures of two subsets of a topological space are equal, are the closures of the images of the two subsets under a continuous map also equal?

Suppose $A$ and also $B$ are parts of a topological room $X$ such that $\newcommand{cl}{\operatorname{cl}}\cl(A) = \cl(B)$. Allow $f\colon X\to Y$ be a continual map of topological rooms. Does that mean that $\cl(f(A)) = \cl(f(B))$?
2022-07-25 17:46:59

Formal proof that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is connected.

Cam any person give me the evidence of: that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is attached.
2022-07-25 17:46:59

Does the 'closure of the interior' equal the 'interior of the closure'?

My solution is no because, $\mathbb{Q}^o = \emptyset$ therefore $\overline{(\mathbb{Q}^o)} = \emptyset$ yet $\overline{\mathbb{Q}} = \mathbb{R}$ therefore $\big(\overline{\mathbb{Q}}\,\big)^o = \mathbb{R}$. Is my instance deal with?
2022-07-25 17:46:51

Completeness of normed spaces

As earlier, I have actually obtained a solution from this website that Bolzano Weierstrass' theory holds true for limited dimensional normed rooms, yet except boundless dimensional rooms. This, specifically = > all limited dim. normed rooms are full (in the feeling that every Cauchy series merges (w.r.t. standard) ). Nonetheless, is it real t...
2022-07-25 17:46:40

The interior of $\mathbb{R} \times \mathbb{Q}$

A question says, find the closure and interior of the sets $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R} \times \mathbb{Q}$. The answers say $\mathbb{R}^2$ and $\emptyset$ respectively for both. Why isn't the interior of $\mathbb{R} \times \mathbb{Q}$, $\mathbb{R} \times \emptyset$ because the interior of $\mathbb{R}$ is $\mathbb{R}$? Does $\...
2022-07-25 17:46:22

About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$

What can we claim concerning the density and also efficiency of the set $X = \mathbb{R}$ with the statistics $$d(x, y) = \frac{|x-y|}{1+|x-y|}\;?$$ I attempted by revealing that $d(x, y)<1$ and also hence it is bounded. Yet just how to show remainder of the points? Can any person aid me? Many thanks
2022-07-25 17:46:18

Given $B \subseteq \overline{A}$ how to show that every open set meeting $B$ also meets $A$.

I am trying to understand the following proof from my lecture notes for a proposition: Let $A$ be a connected subset of a topological space $X$ and suppose $A \subseteq B \subseteq \overline{A}$. Then B is connected. The proof is as follows: If not, then there exists open sets $U, V \subseteq X$ such that $B \cap U$ and $B \cap V$ are disjoint n...
2022-07-25 17:46:10

The topology on $\mathbb{R}$ with sub-basis consisting of all half open intervals $[a,b)$.

Let $\tau$ be to topology on $\mathbb{R}$ with sub-basis consisting of all half open intervals $[a,b)$. How would you find the closure of $(0,1)$ in $\tau$? I'm trying to find the smallest closed set containing $(0,1)$ in that topology but then I realised I don't fully understand what an 'open' interval is. Is an open interval in this topology...
2022-07-25 17:46:03

Condition for an interval being contained in a subset of $\mathbb{R}$

I'd like some input on this problem. It's a different sort from what I've done before, and it's that sort of problem that (I think) feels so nicely intuitive that it's hard to decide if my proof is rigorous or not. How could it be improved? "Let $x$ be a real number, $A$ a subset of $\mathbb{R}$, and $\varepsilon$ a positive number. Prov...
2022-07-25 17:44:47

Example of Baire Space

Can any person provide an instance of a Baire Space, that is neither in your area portable neither metrizable. I would certainly be gratefull additionally for some referrals.
2022-07-25 17:44:47

The closure and interior of $(0,1)$ on $\mathbb{R}$ under the Zariski topology.

On $\mathbb{R}$ under the Zariski topology I don't understand why the closure of $(0,1)$ is $\mathbb{R}$? Clearly this is the only closed set containing $(0,1)$ but I thought we're looking for an open set containing $(0,1)$ under the definition of closure. Also, I don't understand why the interior of $(0,1)$ is $\emptyset$. Under the definition...
2022-07-25 17:43:37

Suspension of the $(-1)$-sphere

I have an uncertainty concerning the building and construction of the suspension of the round of measurement - 1, that is the vacant set. In the adhering to photo I attempt to clarify the suspension of the vacant set: My uncertainty is that I assume that the "part" of the vacant set that result of the item of among the endpoints of ...
2022-07-25 17:42:24

a question on Lindelöf spaces

Let $X=\{(x,y)\in {\Bbb R}^2:y>0\}$ is the subspace of ${\Bbb R}^2$ with the common geography, after that it is still Lindelöf? Otherwise, with which geography can $X$ be made to be Lindelöf?
2022-07-25 17:22:24

Does such subset have a nonempty interior in $\mathbb{R}$?

Let $A$ is a subset of $\mathbb{R}$ and the cardinality of $A$ is $2^\omega$. The question is this: Does the closure of $A$ in $\mathbb{R}$ have a nonempty interior in $\mathbb{R}$? Added: Thank you for helps. In fact, I can't understand the proof of the lemma 2.11 in this . In line 7 of the proof, I don't know why $cl_R(A_{N,M})$ has a non-empt...
2022-07-25 17:22:10

Is $M=\{(x,y)\in (0,\infty )\times\mathbb{R} : y=\sin(\frac{1}{x}) \}$ a closed set in space $((0,\infty )\times\mathbb{R} ,\rho_{e})$?

Is $M=\left \{(x,y)\in (0,\infty )\times\mathbb{R} : y=\sin(\frac{1}{x}) \right \}$ a closed set in space $((0,\infty )\times\mathbb{R} ,\rho_{e})$, $\rho_{e}$ - Euclidean metric ? I think that open set could be built "around" function $y=\sin(\frac{1}{x})$ on domain $(a,\infty)$ where $a>0$ with sum of open balls, but I don't...
2022-07-25 17:21:35

Equivalence of three properties of a metric space.

Another question about the convergence notes by Dr. Pete Clark: (I'm almost at the filters chapter! Getting very excited now!) On page 15, Proposition 4.6 states that for the following three properties of a topological space $X$, $(i)$ $X$ has a countable base. $(ii)$ $X$ is separable. $(iii)$ $X$ is Lindelof (every open cover admits a coun...
2022-07-25 17:20:33

Lindelöf space, closed instead of open.

I recognize that every subspace of $R^k$ is , i.e.: If G is a subspace of $R^k$, after that any kind of open treatment of G has a countable below - covering. I was assuming whether it holds true that, offered G a subspace of $R^k$ any kind of shut treatment of G (covered by shut boxes, (coordinate of each measurement has the kind [a, b], $(-\in...
2022-07-25 17:19:34