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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165 views

Is cantor set homeomorphic to the unit interval?

Can anyone help me with this question? Is cantor set homeomorphic to the unit interval? I (think that I) can see that there is an $f: C \rightarrow [0,1]_{inf}$ which is a surjective bijection ...
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1answer
200 views

Path connectedness is a topological invariant?

$\textbf{PROBLEM}$ Path-connectedness is a topological invariant MY try: we can show that the image of a path connected space $X$ under a continuous mapping is path connected Suppose $X$ is ...
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1answer
134 views

Is each space filling curve everywhere self-intersecting?

Consider a continuous surjection $f:[0,1]\to [0,1]\times[0,1]$. Is $$\{x:\exists(t_1\not=t_2) f(t_1)=f(t_2)=x\}=[0,1]\times[0,1]?$$
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2answers
130 views

Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces?

Let $(X,x)$ and $(Y,y)$ be path-connected pointed topological spaces. Is it true that the statement ''$X$ and $Y$ are homotopy equivalent'' implies ''$(X,x)$ and $(Y,y)$ are homotopy equivalent as ...
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1answer
300 views

If each component of a Cartesian product is homeomorphic to another space, are the Cartesian products homeomorphic

Assume we are given a space $A$ with a metric $d$. Assume $A = A_1 \times A_2 \times A_3 \cdots$, ie. $A$ is a Cartesian product of spaces $A_i$, where $i \in I$. $I$ is countable or countably ...
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1answer
390 views

Complement is connected iff Connected components are Simply Connected

Let $G$ be an open subset of $\mathbb{C}$. Prove that $(\mathbb{C}\cup \{ \infty\})-G$ is connected if and only if every connected component of $G$ is simply connected.
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1answer
202 views

If a collection of sets is a subbase for a topology $\tau_0$ and a base for a topology $\tau_1$, can we conclude $\tau_0 = \tau_1$?

Let $X$ denote a set, and let $\mathcal{O}$ denote a collection of subsets of $X$. Then $\mathcal{O}$ is necessarily a subbase of a unique topology, call it $\tau_0$. And it may or may not be the ...
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80 views

Quotients and products

From the space of real numbers, form the identification space $\Bbb R/\sim$ by identifying $\frac1n\sim~n$, $\forall n$. Describe a typical neighborhood of $0$ on the space $\Bbb R/\sim$. Look at the ...
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147 views

Closed mapping in Hausdorff space.

Consider $(X, \mathcal{T})$. Suppose $X$ is compact, $(Y, \mathcal{T}_Y)$ is Hausdorff. How do we show that the projection map $f: X \times Y \rightarrow Y$ is a closed mapping.
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89 views

A question on the regular space

Here is an exercise: Show that if $A$ and $B$ are disjoint closed subsets of a regualr space $X$ which both have the Lindelof property, then there exist open sets $U, V \subset X$ such that $A ...
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1answer
162 views

How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
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634 views

How to show that $[0,1]^{\omega}$ is not locally compact in the uniform topology?

On page 186 of Munkres' Topology Show that $[0,1]^{\omega}$ is not locally compact in the uniform topology? Uniform topology is defined as topology induced by uniform metric $p$ which is stated ...
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1answer
233 views

where can i find this A.H. Stone's theorem proof?

can someone tell me where can i find a proof of the following theorem (by A.H.Stone) : "an uncountable product of Hausdorff non-compact spaces is never normal " ? thanks in advance !
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4answers
640 views

Intuitive significance open sets (and software for learning topology?)

I have just started to learn topology and I referred to some books and online lectures. The problem is that they all talk the same things and are missing the same things. I want to know "what is the ...
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1answer
107 views

Does “regular” implies collectionwise hausdorff?

Does "regular" implies collectionwise hausdorff? A topological space is said to be collectionwise Hausdorff if given any closed discrete collection of points in the topological space, there are ...
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1answer
343 views

A universal property for the subspace topology

Let $X$ be topological space and $Y$ be a subset of $X$ with $i\colon Y\to X$ the inclusion map. Show that the induced topology of $Y$ is characterized by the following property: A function $f\colon Z ...
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1answer
110 views

Questions for understanding fiber bundle definition

From Wikipedia: A fiber bundle consists of the data $(E, B, π, F)$, where $E, B, $and $F$ are topological spaces and $π : E → B$ is a continuous surjection satisfying a local triviality ...
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492 views

Does a dense $G_\delta$ subset of a complete metric space without isolated points contain a perfect set?

Let $(X,d)$ be a complete metric space without isolated points. Is it true that each dense $G_\delta$ subset of $X$ contains a nonempty perfect set (i.e. closed without isolated points)? Thanks.
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635 views

Show that this countable collection is a basis for $\mathbb R^2$

Show that the countable collection of rectangles $\{ (a,b)\times (c,d) \mid a<b \text{ and } c<d, \text{ and } a,b,c,d \text{ are rational} \}$ is a topological basis for $\mathbb{R^2}$. ...
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2answers
194 views

Four Color Theorems: Graphs vs. Maps [closed]

This question has changed dramatically from its original form. Please See the improved question. ORIGINAL QUESTION: There are two variants of the four color theorem that are commonly cited: (4CTG): ...
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2answers
132 views

Proving an equivalence between equalities

I was reading a textbook about topology and I found this proposition : Let $F$ be a sub-topological space of $E$ and let $A \subset F$. Then $A$ is closed in $F$ if and only if it exsists $B$ closed ...
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80 views

complete metric space $X$ and nested sequence of closed sets $A_m \subset X$ where $\bigcap_{n=1}^\infty = \emptyset$ [duplicate]

What is an example of a complete metric space $X$ and a nested sequence of closed sets $A_m \subset X$ such that $\bigcap_{n=1}^\infty A_m = \emptyset$? My analysis professor mentioned this in office ...
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1answer
136 views

How to prove the Cone is contractible?

Let $\sim$ be an equivalence relation on a topological space $X$ with a subset $A ⊂ X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then define $X/A$ ...
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73 views

Slice at a point of a topological space

The definition is from the following link -Slice at a point of a topological space Let $G$ be a topological transformation group of a Hausdorff space $X$. A subspace $S$ of is called a slice at a ...
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111 views

Constructing a circle from a square [duplicate]

I have seen a [picture like this] several times: featuring a "troll proof" that $\pi=4$. Obviously the construction does not yield a circle, starting from a square, but how to rigorously and ...
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3answers
229 views

Show that an “open square” is an open set: Show that {(x,y) in R2 such that -1<x<1 and -1<y<1. } is an open set.

How do I prove that an "open" square, centered in the origin is in fact an open set? I've already have this geometrical argument: Let $S$ denote the square. Suppose $(x,y) \in S$. Let $\delta = \min ...
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0answers
55 views

What is this space homeomorphic/homotopy equivalent to?

Let $X \subset \mathbb{C}^2$ be given by the equation $$|z|^2 + |w|^2=1$$ and let $A \subset X$ be given by $|z|=1,\ w=0$. This question requires me to find the relative homology groups $H_n(X,A)$, ...
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1answer
37 views

more about locally closedness

"the union of a closed set & an open one is locally closed." i think this statement is false in general but i have no counterexample. does anybody have? in what conditions that holds? explain ...
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51 views

Question about the proof of $S^3/\mathbb{Z}_2 \cong SO(3)$

I'm trying to show $S^3/\mathbb{Z}_2 \cong SO(3)$ completely rigorously. For that purpose I considered three-sphere $S^3$ as a subspace of the ring of quaternions $\mathbb{H}$ and looked into the map ...
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95 views

If $f :\mathbb{R}\to\mathbb{R}$ is measurable, then $E = \{x: f(x) \geq 3\}$ is measurable

Prove: Suppose $f : \mathbb{R}\to\mathbb{R}$ where $f$ is measurable and $E = \{x: f(x) \geq 3\}$. Show $E$ is measurable. I saw this statement while reading in a paper and thought this might ...
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175 views

Show the $\operatorname{int}(A)$ is open.

So we want to show that the interior of any set $A$ is open. We will denote $\operatorname{int}(A)$ as the interior of $A$ which is the set of all interior points of $A$. I know in order to prove ...
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5answers
319 views

Path connected in the set of complex numbers

may i ask for a little help about a proof i have to show. Let $U\subset \mathbb{C}$ be an open and path connected set. Show that if $W\subset U$ closed and open not empty subset, then $U = W$. Thank ...
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1answer
579 views

Extension of a Uniformly Continuous Function between Metric Spaces

Let $(X,d)$ and $(Y,d')$ be metric spaces with $(Y,d')$ complete. Let $A\subseteq X$. I need to show that if $f:A\to Y$ is uniformly continuous, then $f$ can be uniquely extended to $\bar{A}$ ...
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1answer
666 views

Compact metrizable space has a countable basis (Munkres Topology)

Let X be a compact metrizable space. Would you help me to prove that X has a countable basis. Thanks.
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1answer
209 views

Products of quotient topology same as quotient of product topology

Let $X$ be a topological space, $p:X\to Y$ be a quotient map, and $q:X\times X\to Y\times Y$ be the quotient map defined by $q(x,y)=(p(x),p(y))$. Prove that the topologies on $Y$ is the same as the ...
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4answers
343 views

Intersection of compact and discrete subsets

I have difficulties with a rather trivial topological question: A is a discrete subset of $\mathbb{C}$ (complex numbers) and B a compact subset of $\mathbb{C}$. Why is $A \cap B$ finite? I can see ...
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1answer
366 views

(ZF) Every nonempty perfect set in $\mathbb{R}^k$ is uncountable.

This is the part of proof in Rudin PMA p.41 Let $P(\subset \mathbb{R})$ be a perfect set. Since $P$ has limit points, $P$must be infinite. Suppose that $P$ is countable. Then, we can denote the ...
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2answers
863 views

In which topological spaces is every singleton set a zero set?

The title question says it all: if $X$ is a topological space, then a subset $Z$ of $X$ is a zero set if there is a continuous function $f: X \rightarrow \mathbb{R}$ with $Z = f^{-1}(0)$. Now I ...
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1answer
326 views

Continuity of a function to the integers

I am trying to prove that in $\mathbb{Z}$ with co-finite topology the only path-connected components are the singletons. (I reckon that) showing that "if a function $\gamma : [0,1] \to ...
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3answers
986 views

Is a countable, totally disconnected Hausdorff space necessarily totally separated? How about zero-dimensional?

I'm starting to feel a little bad about using this website as my own personal counterexample generator, but here I go again... Terminology: Let's call a space zero-dimensional if it is $T_0$ and ...
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3answers
394 views

How can I prove that this function is continuous at $0$?

The classical example of a functions with only one point of continuity is $$ f(x) = \begin{cases} x & \text{if } x \in \mathbf{Q}, \\ 0 & \text{otherwise}. \end{cases} $$ I only want to ...
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535 views

When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$?

Weyl's criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if $$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$ for ever non-zero integer ...
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1answer
289 views

Extending open maps to Stone-Čech compactifications

Let $X$ be a Čech-complete space, and $Y$ a paracompact space. Suppose $f\colon X\to Y$ is a continuous and open surjection. Since $Y$ is completely regular we have that $\beta(Y)$ is homeomorphic to ...
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278 views

Spaces where all compact subsets are closed

All compact subsets of a Hausdorff space are closed and there are T$_1$ spaces (also T$_1$ sober spaces) with non-closed compact subspaces. So I looking for something in between. Is there a ...
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2answers
273 views

Connectedness of a certain subset of the plane

Let $U$ be an open and connected subspace of the Euclidean plane $\mathbb{R}^2$ and $A\subseteq U$ a subspace which is homeomorphic to the closed unit interval. Is $U\setminus A$ necessarily ...
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2answers
249 views

About the limit of a uniformly converging function sequence

Let $\phi_n\colon [a,b]\to \mathbb R$ be a sequence of continuous functions. Assuming there is an $A\subset [a,b]$ such that $\phi_n|_A$ converges uniformly to $\phi\colon A\to \mathbb R$, I have ...
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1answer
188 views

Is a perfect set a boundary?

In a topological space, is a perfect set (i.e. closed with no isolated points) always the boundary of some set?
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1k views

Compact open sets which are not closed.

Can a nonclosed open subset of a $T_1$ topological space be compact? I mean an open compact set which is not clopen.
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177 views

What conditions are sufficient for “Basically disconnectedness implies Extremally disconnectedness”?

Recall the definition of basically disconnected: A space is basically disconnected if every cozero-set has an open closure. There exists a Basically disconnected space which is not extremally ...
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136 views

$f: \mathbb{R}^2 \to \mathbb{R}$ a continuous open map, show that for each $x \in \text{range}(f)$, $f^{-1}(x)$ is always uncountable.

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuous open map. Show that in fact for each $x \in \text{range}(f)$, $f^{-1}(x)$ is always uncountable. I know that if this was simply a projection onto ...