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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466 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 ...
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1answer
38 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 ...
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1answer
22 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 $...
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1answer
86 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'')$ ), ...
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233 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.
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1answer
93 views

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

Let $U\subseteq \mathbb{R}^2$ be an open set in the standard topology. And let $V\subseteq U$ be a discrete sub-set. Is $V$ necessarily countable? how can I prove it?
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0answers
58 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 ...
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2answers
254 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|| & \text{if} \; \; x,y,P \; \; \text{are collinear,}\\ || x - P|| + ||y-P||& \;\;\;\; \text{...
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1answer
17 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 ...
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1answer
17 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 ...
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3answers
136 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 $\mathbb{R}$ 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?
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1answer
48 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
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1answer
51 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 $[0,1]...
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1answer
126 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
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1answer
65 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 ...
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2answers
167 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 ...
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19 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 ...
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1answer
40 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 ...
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175 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 ...
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106 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 ...
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0answers
24 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 ...
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48 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 showing ...
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72 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 ...
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1answer
27 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 ...
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1answer
399 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
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1answer
31 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 $f : \mathbb{R}^2 \to \mathbb{R}^3$, $f(x,y) = (x,y,\sin(1))$ ...
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1answer
316 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 ...
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1answer
46 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 ...
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2answers
175 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 ...
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2answers
44 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 '...
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66 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 ...
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3answers
212 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 {$K_\...
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1answer
101 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. ...
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25 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 ...
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22 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 = \...
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2answers
269 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 ...
3
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1answer
56 views

Closed sets in a subspace are formed by intersecting the subspace with closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
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1answer
43 views

Isometric Operators: Common Core

Given a Hilbert or Banach space $\mathcal{H}$. Consider two closed operators $S:\mathcal{D}(S)\to\mathcal{H}$ and $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose they're isometric on a common core $\...
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2answers
365 views

When is the free loop space simply connected?

I am not sure if there is an obvious answer to this, but this has been bothering me. Let $X$ be a topological space. When is the free loop space, $LX$, simply connected? Correct me if I'm wrong, but ...
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1answer
65 views

Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
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1answer
97 views

MAth proof questions Open closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
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318 views

Open/closeness of subsets of natural numbers

So I've just started reading about neighbourhood and Hausdorff space. It makes me wonder if $(\mathbb{N},\mathcal{P}(\mathbb{N}))$ is Hausdorff and why, and are sets in $\mathbb{N}$ open or closed or?
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1answer
66 views

Limit points Topology

I'm trying to prove the following: Th: A subset of a topological space is closed iff it contains all of its limit points. Defn of a limit point of a subset $A$ is the following: $p \in X$ is a limit ...
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2answers
54 views

equivalent characterisation of simply connect spaces

I want to prove the following: Let $X$ be path connected space, $S^{1}$ the $1$-sphere and $D^{2}$ the unit circle. Following are equivalent: i)X is simply connected. ii) If $f:S^{1} \to X$ is ...
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2answers
109 views

The Fundamental Group - An explicit homotopy between $(f \circ g) \circ h$ and $f \circ (g \circ h)$

I'm wondering if anyone can help me to understand a proof that the fundamental group is in fact a group. I am looking at the proof on page 3 of this document. I understand everything, although I am ...
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98 views

Lattice Version of Stone-Weierstrass

I've been reviewing Stone-Weierstrass theoerem. While reading the wikipedia page I read the following version of the theorem: Suppose $X$ is a compact Hausdorff space with at least two points and $L$ ...
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3answers
127 views

Topology-Open Sets of a Metric Space

Let $(X_i,d_i), i=1,2,\dots,n$ be metric spaces. Let $X=\prod_{i=1}^{n}X_i$ and let $(X,d)$ be the metric space defined in the standard manner. For $i=1,2,\dots,n$, let $O_i$ be an open subset of $X_i$...
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1answer
266 views

Tietze Extension Theorem

I saw Tietze extension theorem. Since its proof is non-trivial, I tried whether we can clarify it intuitively for functions of one real variable. So, in this special case, I am trying to prove that if ...
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3answers
207 views

Is the plane minus the integer lattice homeomorphic to the plane minus the integers?

The question, more rigorously posed, is: Is $\Bbb R^2-\Bbb Z^2$ homeomorphic to $\Bbb R^2-\Bbb Z\times\{0\}$? This question has been bugging me in the back of my head for a while now. Sometimes, ...
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1answer
61 views

determine if set is open or closed

I have to determine whether the sset {1,2,3} is open or closed. I have never done these types of questions before but this is what I did (on pic). please can I have some feedback if I have done it ...