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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1answer
579 views

Prove the image of separable space under continuous function is separable.

Let $f:X\to Y$ be continuous. Show that if $X$ has a countable dense subset, then $f(X)$ satisfies the same condition.
2
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1answer
76 views

Hausdorff space and disjoint open sets

Let $X$ be a Hausdorff space and $x_1,...,x_n$ points of $X$. Then there exist open pairwise disjoint sets $V_1,...,V_n$ such that $x_i$ is in $V_i$ for every $i$. How can I prove this?
2
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1answer
59 views

If a product is normal, are all of its partial products also normal?

It seems like this should be true, but I can't find the right argument. Thanks. Edit: How about 1) Take two disjoint closed sets in a partial product. 2) Extend them trivially to the entire ...
2
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1answer
268 views

Quotient space and Retractions

I'm trying to learn something about topology and category theory. Let us consider the category of compact Polish spaces. The category contains all quotients of all objects (wikipedia) For an ...
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1answer
213 views

Closed Convex sets of $\mathbb{R^2}$

Can some one please list the closed convex sets in $\mathbb{R^2}$ up to homeomorphism. How many of them are compact
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3answers
140 views

If $x$ is not in $A$, a closed set in a Metric space then $d(x,A)>0$

If $A$ is a closed in a metric space $(X,d)$ with $x\notin A$, I need to show that $d(x,A)>0$. Now assume $d(x,A)=0$ then $\exists x_n\in A $ s.t.$d(x_n,A)=0$ then there is a sequence in $A$ s.t. ...
2
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3answers
149 views

A question on $P$-space

A space $X$ called $P$-space if every $G_\delta$ subset of $X$ is open in $X$. Every discrete space, I believe, is $P$-space. My question is this: Could someone offer some other classical examples of ...
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2answers
1k views

How to prove that the uniform topology is different from both the product and the box topology?

Let $J$ be an arbitrary index set. Then how to prove that the uniform topology on the Cartesian product $\mathbf{R}^J$ of the set $\mathbf{R}$ of real numbers with itself is different from both the ...
2
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2answers
211 views

Using functions to separate a compact set from a closed set in a completely regular space

Suppose $X$ is completely regular, $K\subset X$ compact and $C\subset X$ closed such that $K\cap C$ empty. Prove that there exists $f_x\in C(X,[0,1])$ such that $f_x(x)=1$ and $f_x=1$ on $C$ for ...
2
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1answer
326 views

Why is this not a proof of Invariance of Domain?

We know that if $f:K \to X$ is continuous and injective, $K$ is compact, and $X$ is Hausdorff, then $f$ is a homeomorphism $K \cong f(K)$. So suppose $f:U \to \mathbb{R}^n$ is continuous and ...
2
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1answer
703 views

Show that this set is connected but not path connected

I wonder whether someone can do me a favor in proving the following union of the sets $C$ is connected but not path connected: $[0, 1] \times \{0\}$$\{1/n\}\times [0, 1]$$\{(0, 1)\}$ For now, I have ...
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2answers
603 views

Homotopy equivalence between n times punctured plane and…

how to prove that $\mathbb{R}^2$ without $n$ points is homotopy equivalent to $S^1 \vee ... \vee S^1 $ which means a bouquet of $n$ circles? It's easy for $n=1$ and $n=2$ but how to generalize it? By ...
2
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1answer
913 views

Closed subset of complete metric space…don't understand last part of theorem.

A closed subset of a complete metric space is a complete subspace. Proof. Let $S$ be a closed subspace of a complete metric space X. Let $(x_n)$ be a Cauchy sequence in $S$. Then $(x_n)$ is a ...
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4answers
401 views

Sum of closed convex set and unit ball in normed space

Let $(X, \|.\|)$be a real normed space. Let $A$ be a closed convex suset of $X$ and $\mathbb{B}$ a unit ball in X, i.e. $$ \mathbb{B}=\{x\in X: \|x\|\leq 1\}. $$ I would like to ask whether ...
2
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1answer
156 views

How do I prove that $\mathbb CP^n$ is a 2n-manifold?

I'm struggling to prove that $\mathbb CP^n$ is 2n-manifold. We can defined the $\mathbb CP^n$ as the equivalence relation $(z_1,z_1,...,z_{n+1})\sim(w_1,w_1,...,w_{n+1})$ iff $z_i=\lambda w_i$, ...
2
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2answers
181 views

Which functions on N extend uniquely to a continuous function on the Stone-Cech Compactification of N?

The question is exactly the title. Is there a good classification of which functions from $\mathbb{N}$ to $\mathbb{N}$ (or, more generally, from $\mathbb{N}^n$ to $\mathbb{N}$)? Also, what is a good ...
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3answers
2k views

(Un-)Countable union of open sets

Let $A_i$ be open subsets of $\Omega$. Then $A_0 \cap A_1$ and $A_0 \cup A_1$ are open sets as well. Thereby follows, that also $\bigcap_{i=1}^N A_i$ and $\bigcup_{i=1}^N A_i$ are open sets. My ...
2
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1answer
123 views

Relative merits, in ZF(C), of definitions of “topological basis”.

Typical Terminology: A basis $\mathcal{B}$ for a topology on a set $X$ is a set of subsets of $X$ such that (i) for all $x\in X$ there is some $U\in\mathcal{B}$ such that $x\in U$, and (ii) if $x\in ...
2
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1answer
172 views

Irreducible set

Let $X$ be a topological space Let $A$ be the set of all closed, irreducible subsets of $X$ equipped with a topology that contains all sets of the form $V(U)=\{a\in A| a\cap U\neq\emptyset, ...
2
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2answers
595 views

Totally disconnected space

I strongly believe that a totally disconnected Hausdorff space has to have a topology that contains all the singletons. However, I am not sure how to go about proving it. I am especially unsure of how ...
2
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1answer
1k views

Base and subbase of a topology

I'm confused about subbases: the sub in the name suggests that a subbase $S$ is a subset of a base $B$ of a topology $T$. Can there be a topology $T$ such that it is generated by a subbase that is ...
2
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1answer
161 views

Proving a space is Hausdorff, given a specific construction

I'm trying to prove that if in (Haus, U) a morphism $f: X \rightarrow Y$ is not dense, it isn't an epimorphism. To prove this I need the following construction: Suppose $M = \overline{f(X)}$ is ...
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3answers
962 views

Are the rationals a nowhere dense set?

I thought a nowhere dense set was a set where the closure of the complement is the whole space but surely the complement of the rationals is the irrationals who's closure is the reals so surely it's ...
2
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1answer
338 views

Some questions about set closure / Kuratowski closure.

Here is the page that is confusing me: Page 25 of General Topology (Willard) Definition 3.5 If $X$ is a topological space, and $E\subset X$, then the closure of $E$ in $X$ is the set ...
2
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1answer
214 views

separation properties in Hausdorff, compact spaces

Suppose $X$ is a compact Hausdorff topological space, $C\subseteq X$ a closed subset and $x\notin C$ a point. I have to prove that there exists a compact neighborhood of $x$ which is disjoint from ...
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4answers
968 views

Does there any non discrete metric space in which closure of an open ball is not closed ball?

In other words does there exist a non discrete metric space in which Closed balls are subset of the closure of open balls?
2
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0answers
55 views

Is the functor associating a bundle with a structure group to a principal bundle faithful?

Consider a (Cartan) principal G-bundle $\xi: X \to B$, and a left $G$-space $F$. We construct the bundle $\xi[F]: X_F \to B$ associated with $\xi$ with a fiber $F$ as usual. Now for each morphism $(u, ...
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1answer
640 views

The mapping cylinder of CW complex

If $X,Y$ are CW complexes and $f$ a cellular map from $X$ to $Y$, is it true that $M_f$ is a CW complex?
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2answers
88 views

How to prove the group of automorphisms of $S^1$ as a topological group is $\mathbb Z_2$?

The title basically says it all. How does one prove the group of automorphisms of $S^1$ (the unit circle in $\mathbb C$), as a topological group, is $\mathbb Z_2$? I was surprised not to find the ...
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2answers
89 views

Use definitions to show $[0, 1) × [0, 1)$ is neither an open nor closed subset of $\Bbb{R^2}$.

Show, from the definitions of open and closed sets, that when using the standard Euclidean metric, [0, 1) × [0, 1) is neither an open nor closed subset of $\Bbb{R^2}$. From what I understand, a set ...
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2answers
77 views

Uniqueness of Topology on Finite Set

Prove that if $X$ is a finite set with preorder $R$, then exactly one topology $\mathcal{T}$ on $X$ satisfies $\trianglelefteq_{\mathcal{T}} =R$. > Here is the definition the mentioned preorder: ...
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1answer
45 views

What is an example of a nonmetrizable topological space? [duplicate]

Find topological spaces $X$ and $Y$ and a function $f:X\to Y$ which is not continuous, but which has the property that $f(x_n)\to f(x)$ in $Y$ whenever $x_n\to x$ in $X$ I know this is true if $X$ is ...
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0answers
44 views

Find a topology $\mathscr{T}_0$ for which $f,g : ( \mathbb{R}^2, \mathscr{T}_0) → (\mathbb{R}^2, \mathscr{T}_{R}^{c} )$ is continuous.

For $f,g : \mathbb R \rightarrow \mathbb R$, define: $$ x \rightarrow f(x) := \begin{cases} x^2 & \text{for }x \le 1 \\ x+1 & \text{for }x > 1 \end{cases} $$ $$ x \rightarrow g(x) := ...
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2answers
48 views

Proving an operation of interior is a set of open sets.

I am looking at General Topology notes on Alternative Ways of Defining Topology and come up with this questions: If $\iota : \mathscr P (X) \to \mathscr P (X)$ is an operation of interior, then $\tau ...
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0answers
72 views

Topological entropy of isometric extension

L.s., This is a homework question some of my fellow students and I are having great difficulty with. Let $Y,Z$ be compact metric spaces, $X = Y \times Z$, and $\pi$ the projection to $Y$. Denote $h$ ...
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2answers
122 views

Help with a proof that SO(n) is path-connected.

I've found lots of different proofs that SO(n) is path connected, but I'm trying to understand one I found on Stillwell's book "Naive Lie Theory". It's fairly informal and talks about paths in a very ...
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1answer
59 views

Is an algebra the smallest one generated by a certain subset of it?

Let $X$ be a completely regular topological space and let $BC(X)$ denote the space of bounded continuous complex-valued functions on it. Also, let $C(X,[0,1])$ be the set of continuous functions on ...
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1answer
94 views

Finding the closest point in a set to another point in n-dimensional space: efficiently

I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
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1answer
134 views

What is meant by gluing two metric spaces together?

"Gluing" constructions are common in topology: by gluing two disks along their boundaries we get a sphere; by gluing a cylindical "handle" to a sphere we get a torus, and so forth. If the original ...
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1answer
46 views

Closure of a bounded set

I have troubles with proving that the closure of a bounded set is bounded. Can somebody help me out with this problem? Thanks!
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1answer
55 views

Some troubles about topology and definition of a Vector Bundle

Disclaimer: It's heavily related to my old question : Visualizing the Topology of a Vector Bundle but I wanted to open a new question because the former had already got an answer and this time my ...
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0answers
126 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
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2answers
814 views

Proof that a product of two quasi-compact spaces is quasi-compact without Axiom of Choice

A topological space is called quasi-compact if every open cover of it has a finite subcover. Let $X, Y$ be quasi-compact spaces, $Z = X\times Y$. The usual proof that $Z$ is quasi-compact uses a ...
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2answers
137 views

How is $\mathbb R^2\setminus \mathbb Q^2$ path connected?

Prove $(\mathbb R$ x $\mathbb R)-(\mathbb Q$ x$ \mathbb Q)$ is path connected. I know I need to let $(x_0, y_0), (x_1, y_1) \in$$(\mathbb R$ x $\mathbb R)-(\mathbb Q$ x$ \mathbb Q)$ and then consider ...
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1answer
272 views

continuous real valued functions on the ordinal space $[0,\Omega)$

As a continuation to this question. Let $X$ be the ordinal space $[0,\Omega)$, with the order topology, where $\Omega$ is the first uncountable ordinal. Let $f:X \rightarrow \mathbb R$ be a ...
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3answers
144 views

Open and closed maps: What good for?

I'm wondering what open mappings are actually good for (except for inverse becomes continuous)??? My irritation came since, people stress that an open mapping not necessarily preserves closed sets ...
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1answer
174 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
223 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
252 views

$\{A_\alpha\}$ closed, locally finite, $f\restriction_{A_\alpha}$ continuous; show $f$ continuous

Let $\{ A_\alpha \}$ be a collection of subsets of $X$; let $X=\bigcup_\alpha A_\alpha$. Let $f:X\rightarrow Y$; suppose that $f\restriction_{A_\alpha}$, is continuous for each $\alpha$. Show that ...
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1answer
136 views

Is each space filling curve everywhere self-intersecting?

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