Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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3
votes
1answer
19 views

Why is $V(x)\cup(\mathbb{A}^2\setminus V(y))$ not quasi-affine?

I'm having trouble understanding the following situation. Apparently it's not difficult to see the union $V(x)\cup(\mathbb{A}^2\setminus V(y))$ is not a quasi-affine set. Everything is being done ...
1
vote
2answers
65 views

Two continuous functions with connected images

Suppose we have two continuous functions $f(x)$ and $g(x)$. Define $f$ on $[0,1]$ and $g$ on $[1,2]$, such that $f(1)=g(1)$. If we know that $\text {Im} (f(x))$ and $\text{Im} (g(x))$ are connected, ...
1
vote
1answer
19 views

Is this proof of $\overline{\cup}A_\alpha \subset \cup \overline{A_\alpha}$ valid?

This is a problem from my topology homework: Let $x \in \overline{\cup}A_\alpha$. Then for every $U(x)$, $U(x) \cap (\cup A_\alpha) \neq \emptyset \rightarrow U(x) \cap A_\alpha \neq \emptyset$ ...
1
vote
1answer
26 views

Are there actually 4 types of Product Topologies?

We learned about the Product Topology recently in class. So for $X,Y$ topological spaces, then we define $X\times{Y}$ to be the product topology. From what I understand, for $U\subset{X}$ and ...
3
votes
2answers
77 views

Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
2
votes
2answers
86 views

Homeomorphism between plane with different topologies

How would you show that spaces $(\mathbb{R^2},\cal{T}_r)$ and $(\mathbb{R}^2,\cal{T}_b)$, where $\cal{T}_r$ is a topology generated by jungle river metric (here) and equivalently $\cal{T}_b$ is ...
0
votes
0answers
31 views

Hilbert Subspaces: ONB

This might be a duplicate. If so, then please let me know - I will close this thread then. Thanks! Given a Hilbert space $\mathcal{H}$. Consider a dense subspace $\overline{Z}=\mathcal{H}$. Then it ...
0
votes
4answers
55 views

Subset of $\mathbb{R}$ is countable iff it contains no interval?

I was doing some topology homework and used this equivalence to complete a problem, only I'm not completely sure it actually holds. My intuition tells me it's true, but I can't prove it. If it helps, ...
0
votes
0answers
10 views

Patition of unity by real valued continuous function on compact Hausdorff space.

Let $X$ be a compact Hausdorff space and let $\{U_\alpha\}_{a \in A }$ be an open cover of $X$. Show that there exit a finite number of continuous real valued functions $h_1\cdots h_n$ on $X$ with the ...
0
votes
1answer
26 views

Any open set shares boundary with a discrete set

Claim: Let $X$ be a metric space and let $U\subset X$ be open. Then there exists a discrete set $A\subset X$ such that $\partial A = \partial U$. Approach thus far: Since this statement is about ...
2
votes
5answers
38 views

Well-ordered set with greatest element is compact

Let $X$ be a well-ordered set with a greatest element $\alpha$. We consider all sets of the form $]x,y]$ where $y \in X$ and $x$ is either another element of $X$ or the symbol $\leftarrow$ (the ...
0
votes
0answers
29 views

The Complex projective space is homeomorphic to the n-sphere

Ok I have been asked to give as detailed a proof as I can for the following question. Prove that $ \mathbb C\mathbb P^n $ is homeomorphic to $ S^{2n+1} /\sim. $ where for $ z,w \in S^{2n+1} \subset ...
0
votes
1answer
28 views

proving that a set is closed

Let X be compact, metric, connected and locally connected space. And let M$\subseteq$X closed and connected. a,b,p$\in$M are non-cut points of X. I showed that M$\setminus${p} is connected. Now I ...
0
votes
0answers
59 views

Topological properties of these sets

Consider the following sets: $A = \{(x,y) \in \mathbb{R} ^2: \sin(x)\cos(x)=0\}$ $B= \{ (x,y,z) \in \mathbb{R}^3 : |x| + |z| \leq 1, |x| \leq 1\}$ $C=\{(x,y) \in \mathbb{R}^2 : 1 < ...
0
votes
1answer
8 views

$D$ a closed entourage, $K$ compact subset, show that $D[K]$ is closed.

I'm studying for my topology exam and have come across a question that I can't solve. To state the problem more clearly: For $D$ a closed entourage in a uniform space $X$, and $K$ a compact subset of ...
1
vote
2answers
29 views

Find sets of points, where function from one topological space to another is continuous.

We have got two functions : $f(x,y) = (2x,y)$ $g(x,y) = (x+1,y) $ They are transormations from one topological space to another ( from $ (\mathbb{R^2}, \tau')$ to $ (\mathbb{R^2}, \tau'')$ ), ...
0
votes
1answer
23 views

Why is the topology of compactly supported smooth function in $\mathbb R^d$ not first countable?

In other words, given a countable sequence of neighborhoods of $f(x)=0$, how to construct another open neighborhood that doesn't contain any of these neighborhoods? Thanks.
0
votes
0answers
15 views

Existence of half-planes with respect to regular open sets of the Euclidean plane

Let $\langle\mathrm{r}\mathscr{O},\mathord{\subseteq}\rangle$ be the complete Boolean algebra of open domains (regular open sets, these that are equal to the interior of their closure: ...
1
vote
1answer
37 views

discrete subset of $\mathbb{R}^2$

I have U$\subseteq$ $\mathbb{R}^2$ an open set in the standard topology. And I have V$\subseteq$U discrete set. Is V necessarily countable? how can I prove it?
0
votes
0answers
24 views

Textbooks to complete concurrently - Self learning empowerment

A user is completing some year challenge that takes them through $9$ textbooks and they are alternating in author. Algebra - Cohn Analysis - Rudin Topology - Lee Repeat three times. I would like ...
1
vote
0answers
17 views

Separated Spaces and a Partition Differences?

I am just getting a handle on separated definitions from Topology , reading Munkres. So the definition of a separated subsets of a topology, is that they are both disjoint. Further, if each subset ...
1
vote
0answers
20 views

Having difficulties showing the triangle inequality of metric in the plane

Let $P \in \mathbb{R}^2$ and define $$ d(x,y) = \left\{ \begin{array}{lr} ||x-y|| & if \; \; x,y,P \; \; \text{Are Collinear}\\ || x - P|| + ||y-P||& \;\;\;\; otherwise ...
0
votes
1answer
14 views

Define $f(y)=d(x_0,y)$, prove that $f$ is continuous.

Consider a metric space $(X,d)$ and some $x_o \in X$. Define function $f_{x_0}(y)=d(x_0,y), $ which is in $\text{R}$. Show that the function is continuous. Here's what I've tried: According to ...
0
votes
1answer
10 views

Demonstrating the connectedness of the set $A_j = \{(1,0),(0,0)\} \cup \{(x,y) : 0 < y < 1/j\}$

I'm trying to demonstrate the connectedness of the set $A_j = \{(1,0),(0,0)\} \cup \{(x,y) : 0 < y < 1/j\}$. This is for my class in real analysis, so I can't apply concepts that are too ...
2
votes
3answers
75 views

If a set $S\subset\mathbb R$ is not closed, does it contain a convergent sequence with a limit outside of $S$?

Suppose S is a subset of R and that S is not closed. Must it follow that there is a convergent sequence in S that converges to some l not in S?
1
vote
1answer
33 views

Separation and Hausdorff

I am just learning the definitions of a topological space being separated, but what is the relationship between separated topological space and a Hausdorff space? The definition of separated is that ...
2
votes
1answer
37 views

Two topological spaces which imbed in each other and are quotients of each other but not homeomorphic?

Does there exist two topological space $X$ and $Y$ such that $X$ and $Y$ imbed in each other, $X$ is a quotient of $Y$, $Y$ is quotient of $X$, but $X$ and $Y$ are not homeomorphic? The spaces ...
0
votes
1answer
40 views

Is $\mathbb{R}^{[0,1]}$ separable?

I was trying to disprove (or also prove) whether $\mathbb{R}^{[0,1]}$ is separable. My intuition tells me it's a disprove. I thought perhaps proving that $\mathbb{R}^{[0,1]}$ is sequentially compact ...
3
votes
1answer
41 views

Normal Operator: Everywhere defined implies bounded?

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{H}\to\mathcal{H}$. If its domain is the whole Hilbert space then is it necessarily bounded? The point is that I'm trying ...
2
votes
2answers
38 views

Let $L_{n}$ be a line in $\mathbb{R}^2$ for n = 1,2,3… Prove that $\cup_{n=1}^{\infty} L_n \ne \mathbb{R}^2$.

Question: Let $L_{n}$ be a line in $\mathbb{R}^2$ for n = 1,2,3... Prove that $\cup_{n=1}^{\infty} L_n \ne \mathbb{R}^2$. (Rudin) Attempted Answer: My first thought was using the fact that a line ...
1
vote
0answers
10 views

Closed square homemorphic to the surface of a cube?

Is the closed square in $\mathbb{R}^2$, i.e. $[0,1]^2$ homeomorphic to the surface of the cube in $\mathbb{R}^3$? If they are, is there an explicit homeomorphism? I'm looking for something more solid ...
1
vote
1answer
24 views

Show that g*c and c*g are homotopic, where g is a loop and c a constant loop

I've stumbled upon a question which asks me to prove that $f_0 := g*c$ and $f_1 := c*g$ are homotopic. More specifically it wants me to give a 'picture in $I\times I$, a picture in $X$ and an explicit ...
1
vote
2answers
40 views

Accumulation points and closed sets

Denote by $F$ the set of all accumulation points of $(x_{n})$. We define an accumulation point $x \in \mathbb{R}$ if there exists a subsequence $(x_{n_{k}})$ of $(x_{n})$ (being the latter a bounded ...
1
vote
2answers
29 views

Can a dense set contain isolated points?

I was interested in this question, can a dense set contain an isolated point, because I was reading into the lexicographic order topology on the unit square. I read in here that: $S$ is not ...
1
vote
0answers
15 views

Find all regions formed by a set of circles

I was doodling with Python to draw some circles, and I was able to find all intersection points of a set of random circles, yay ! Now I'm stuck on a question, is there a way to find all regions ...
2
votes
0answers
25 views

The plane minus a countable set homeomorphic to the plane minus an uncountable set?

Is it possible that $\Bbb R^2-C$ can be homeomorphic to $\Bbb R^2-U$ where $C$ is countably infinite and $U$ is uncountable? Intuitively I believe the answer is no, but I'm having difficulty ...
4
votes
2answers
52 views

A few questions on the properties of $\mathbb{R} ^ {[0,1]}$

Given the topological space $X=\mathbb{R}^{[0,1]}$ with the product topology, there are several properties regarding to $X$ which I am not sure if are true/false. Is $X$ metrizable? I'm having ...
1
vote
1answer
13 views

Show that $H= \cup _{r \in \mathbb {Q } \cap [0,1 ] }(K+r) $ is bounded, where $K $ is compact .

I want to show that $H= \cup _{r \in \mathbb {Q } \cap [0,1 ] }(K+r) $ is bounded, where $K + r $ denotes the translate and $K $ is compact . This is sort of obvious but I want to construct an ...
3
votes
1answer
34 views

is the lexicographic order topology on the unit square connected/path connected?

I was wondering, given the lexicographic order topology on $S=[0,1] \times [0,1]$, is it connected (and path connected)? I found a reference to Steen's and Seebach's Counterexamples in Topology, and ...
3
votes
1answer
21 views

Determining the connectedness of $\{(x,y,\sin(x^2+y^2)) : x^2+y^2=1\}$

This question is on my exam review sheet. It says we can use the fact that $\{(x,y) : x^2+y^2 = 1\}$ is connected. Am I correct in saying $f : \mathbb{R}^2 \to \mathbb{R}^3$, $f(x,y) = (x,y,\sin(1))$ ...
4
votes
1answer
52 views

Is every linear ordered set normal in its order topology?

I'm trying to prove (or disprove) that every linear ordered set $(X, <_X)$ is normal in its order topology. I was able to prove $(X,<_X)$ is hausdorff, simply by taking two open intervals with ...
2
votes
1answer
23 views

Topology, small detail on a proof. Concept Closure and Adherent

I'm trying to recreate the proof that: If $Y $ is a subset of a metric space $ X $, then the closure of $ Y $ is closed. But I cannot provide a proof of the following: Let $ x \in \overline{\overline ...
2
votes
2answers
43 views

Topologies in a Riemannian Manifold

I'm studying Differential Manifolds using Manfredo do Carmo's Book (Riemannian Geometry) and although I see no mention of this in Do Carmo's book, it's really easy to see a Riemannian Manifold as a ...
3
votes
2answers
27 views

The plane minus the graph of a continuous function consists of two path-connected components?

Let $f:\Bbb R\rightarrow \Bbb R$ be continuous. Show that $\Bbb R^2-\mathrm{graph}(f)$ consists of two path-connected components. I can show that the area 'above' the graph of $f$ and the area ...
2
votes
0answers
30 views

Showing that the image of a polynomial map is not closed

Let $f : \mathbb{C}^3 \rightarrow \mathbb{C}^4$ be defined by $(s, t, u) \rightarrow (st, st^2+(1-s)u, st^3, 1-s)$, where $\mathbb{C}$ denotes the complex numbers. Then for some irreducible ...
2
votes
3answers
34 views

The nonemptiness of the intersection of compact sets such that all finite intersections are nonempty

From Rudin's Principles of Mathematical Analysis: Theorem 2.36: If {$K_\alpha$} is a collection of compact sets of a metric space X such that the intersection of every finite subcollection of ...
2
votes
1answer
28 views

Complement of closed dense set

Let $X$ be a topological space and $C$ be its closed and dense subset. Then is it possible for $X-C$ to be dense in X? I think $C$ doesn't have to be closed, and in that case $X-C$ can be also dense. ...
1
vote
0answers
9 views

Fundamental domain for a $C_2$-action on a Stone space

The following result seems to be true (I can prove it, only quite indirectly): Let $X$ be a Stone space (i.e. a compact totally disconnected Hausdorff space) and $\sigma : X \to X$ be a ...
1
vote
0answers
18 views

Question about Hausdorff spaces and their equivalences [duplicate]

Definition: A topological space $X$ is called Hausdorff space if for each $x_1,x_2 \in X$ (they are distinct) we can always find neighborhoods $U_1,U_2$ of $x_1,x_2$ such that $U_1 \cap U_2 = ...
2
votes
2answers
39 views

Show that the set is compact using the definition

The set in question is $\{0\}\cup \{1,\frac12,\frac13,\ldots,\frac1n,\ldots\}$ (for $n\in\mathbb N$). Okay, so for a set to be compact, every open cover of it must be able to be broken down into a ...