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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The $n$-disk $D^n$ quotiented by its boundary $S^{n-1}$ gives $S^n$

Define $D^n = \{ x \in \mathbb{R}^n : |x| \leq 1 \}$. By identifying all the points of $S^{n-1}$ we get a topological space which is intuitively homeomorphic to $S^n$. If $n = 2$, this can be ...
5
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1answer
2k views

When is the union of topologies a topology?

The union of two topologies on some set may or may not be a topology. When is it a topology?
5
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1answer
303 views

An exercise about finite intersection property in $T_1$ space

Let $X$ be a $T_1$ space. Let $\mathfrak {D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. Show that there is at most one point belonging to $\...
5
votes
0answers
169 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have $$d(x,...
5
votes
1answer
97 views

Surjectivity of a continuous map between $\mathbb{R}^d$s

Let $f:\mathbb{R}^{d}\to\mathbb{R}^d$ be a continuous map. Show that if $\displaystyle\sup_{x\in\mathbb{R}^d}|f(x)-x|<\infty$, then $f$ is surjective. I encountered this problem more than 3 ...
4
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3answers
1k views

What is the interior of a singleton?

I was wondering what can be said about the interior of $\{{4}\}$, the empty set? The interior of a set $A$ is the largest open set contained by $A$. Hence, if the set at hand is a singleton, then isn'...
4
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2answers
356 views

Proof of $(0,1)$ is not compact with usual metric.

In the proof we say $\left\{\left(\frac1n,1\right):n\geq 1\right\}$ is an infinite cover with no finite subcover. But, $(0,1)$ set also belongs to cover mentioned above. We can say $\{(0,1)\}$ is a ...
4
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2answers
588 views

Spaces with the property: Uniformly continuous equals continuous

I found a nice book about functional analysis with a nice theorem in it: Continuity at 0 is equal to Lipschitz continuous for linear maps in normed spaces. This fact inspires me to ask: Are there ...
4
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2answers
407 views

Second Countability of Euclidean Spaces

Sorry I know this is a stupid question. However I got stuck on this for quite a while. I'm trying to prove that Euclidean spaces have a countable base, which can be constructed by taking all the open ...
4
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1answer
2k views

Orthogonal matrices forms a compact set [duplicate]

Could someone help me to prove that the set of all $n\times n$ orthogonal matrices is a compact subset of $\mathbb{R}^{n^2}$. I don't know how it can be done. Thanks.
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3answers
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$[0,1]^{\mathbb{N}}$ with respect to the box topology is not compact

could anyone help to show that $[0,1]^{\mathbb{N}}$ with respect to the box topology is not compact? Thank you!
4
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1answer
162 views

Every open subset of $\mathbb{R}$ can be expressed uniquely as a disjoint union of open intervals. Does this generalize to $\mathbb{R}^n$?

I know that every open subset of $\mathbb{R}$ can be expressed uniquely as a disjoint union of open intervals. Further, only countably many intervals feature in any such decomposition. Supposing we ...
4
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2answers
2k views

Measure on topological spaces

So, given a topological space $S$ we can construct its Borel sigma-algebra $\mathcal{B}(S)$. Does it mean that we can construct a measure $\mu$ on this sigma-algebra as well? Say, discrete topology on ...
4
votes
1answer
293 views

Tychonoff theorem (1/2)

I was trying to prove Tychonoff theorem. First I used (which I showed also): The following are equivalent (a) $X$ is compact (b) every filter of closed set $F$ on $X$ has non-empty ...
4
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2answers
292 views

Compact metric space group $Iso(X,d)$ is also compact

Could you tell me how to prove that if metric space $(X,d)$ is compact, then the group $Iso(X,d)$ is also compact? The group $Iso(X,d)$ is considered with topology determined by a metric $\rho$ on $...
4
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3answers
798 views

Prove that $\Bbb R^2 - \{0\}$ is homeomorphic to $S^1 \times \Bbb R$.

No idea where to even begin. There is a hint: this requires construction of an explicit function.
3
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1answer
561 views

Is the product of Polish spaces Polish?

If $T_t=(E_t,\tau_t)$ are Polish, $t\in I$, is $T^I:=(\prod_{t\in I}E_t,\prod_{t\in I}\tau_t)$ Polish? ($\prod_{t\in I}\tau_t$ being the product topology.) If this is not the case, what extra ...
3
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1answer
629 views

Finite unions and intersections $F_\sigma$ and $G_\delta$ sets

Why is the intersection of finitely many $F_\sigma$ sets an $F_\sigma$ set, and the union of finitely many $G_\delta$ sets a $G_\delta$ set?
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2answers
82 views

Continuous mapping from open set to open set

Suppose we have two open and bounded sets, $\Omega_1,\Omega_2 \in \mathbb{R}^2$. Is there a continuous function $\textbf{f}$ mapping $\Omega_1$ onto $\Omega_2$? \begin{align*} \Omega_1 & = \{(x,...
3
votes
3answers
287 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 ...
3
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2answers
84 views

Lie group step in proof

Let $X_e,Y_e \in T_eG$ be vectors and $G = GL(n).$ Then the right translation is given by $Y_g = Y_e g$ and $X_g = X_e g.$ Now, I have a proof showing that $[X_e,Y_e] \in T_eG$ is the element ...
3
votes
1answer
1k views

Proof that Sorgenfrey plane is not normal using points x × (-x)

I'm making Exercise 9 of paragraph 31 in Munkres, which is a proof that the Sorgenfrey Plane $\mathbb{R}_l^2$ is not normal. I'm having trouble on part c of the question. The full question is: Let $A$...
3
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2answers
534 views

Open Dense Subset of $M_n(\mathbb{R})$

Well, I know the fact that $GL_n(\mathbb{R})$ is open set in $M_n(\mathbb{R})$, how to show that it is dense also? Well I thought like this: If $A\in M_n(\mathbb{R})$ and If $\lambda_1,\dots,\lambda_n$...
3
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1answer
724 views

Continuous image of the intersection of decreasing sets in a compact space

Suppose $B_{\epsilon}$ are closed subsets of a compact space and $B_{\epsilon} \supset B_{\epsilon'} \quad \forall \epsilon > \epsilon'$. Furthermore, $B_0 = \bigcap_{\epsilon>0} B_{\epsilon}$. ...
2
votes
2answers
95 views

Why $Z_p$ is closed.

Let $A_n=\mathbb{Z}/p^n\mathbb{Z}$ be a ring and $p$ is prime, $\phi_n: A_n\rightarrow A_{n-1}$ be a natural homomorphism (Elements of $A_{n}$ define in an obvious way elements of $A_{n-1}$). Define $...
2
votes
1answer
1k views

boundary of the boundary of a set is empty

I am learning some stuff about the interior, closure and boundary of sets $A\subset\mathbb R^n$ and I am wondering about the following: 1) $\partial\partial A=\partial A$ ? 2) $\partial\partial\...
2
votes
2answers
1k 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 $\bar{A}$ ...
2
votes
2answers
1k views

What does order topology over Ordinal numbers look like, and how does it work?

Space of ordinal numbers are one of the favorite examples of my professor in general topology. I quite understand the idea at the base of ordinal numbers (few things or nothing about the concept of ...
2
votes
1answer
230 views

Is there a topological space which is star compact but not star countable?

A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. A ...
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2answers
66 views

Examples of a quotient map not closed and quotient space not Hausdorff

Is there any example of a closed relation $\sim$ on a Hausdorff space $X$ such that $X/\sim$ is not Hausdorff? Also, is there any example of a closed relation ~ on a Hausdorff space $X$ such that a ...
16
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2answers
405 views

A sort of inverse question in topology

Given topological spaces $X$ and $Y$, we often consider the collection of continuous functions, $f: X \rightarrow Y$. My question is, given two sets $X$ and $Y$, and a sub-collection $\{g_{i}\}$ of ...
14
votes
5answers
424 views

Compactness in $\mathbb{Q}$

Proving that $[0,1]\subset\mathbb{R}$ is compact you make use of completeness and this is a fundamental step in order to characterize compact subsets of $\mathbb{R}$. Trying to state an analogous ...
14
votes
1answer
134 views

Exists homeomorphism which carries each fiber isomorphically to itself, composition?

Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see that there exists a homeomorphism $f: E(\xi) \to E(\xi)$ which carries each fiber isomorphically ...
13
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9answers
1k views

Why did we define the concept of continuity originally, and why it is defined the way it is?

The concept of continuity is a very important idea in topology. Though I am using it all the time, but indeed I don't know what is the original purpose for us to define this concept. And I also don't ...
13
votes
1answer
324 views

$\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic

Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
12
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4answers
8k views

Cauchy sequence is convergent iff it has a convergent subsequence

Prove that if $\left ( x_{n} \right )$ is a Cauchy sequence in a metric space X then $\left ( x_{n} \right )$ is convergent if and only if $\left ( x_{n} \right )$ has a convergent subsequence. Note: ...
12
votes
1answer
318 views

On different definitions of neighbourhood.

I am going through the basics of topology, mainly to refresh them. I had taken a course some years ago but never used topology actively. So I am reading Munkres's Topology. I have noticed that he ...
12
votes
2answers
318 views

Is every suborder of $\mathbb{R}$ homeomorphic to some subspace of $\mathbb{R}$?

Let $X \subset \mathbb{R}$ and give $X$ its order topology. (When) is it true that $X$ is homeomorphic to some subspace of $Y \subset \mathbb{R}$? Example: Let $X = [0,1) \cup \{2\} \cup (3,4]$, ...
11
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2answers
412 views

A set which is neither meagre nor comeagre in any interval.

As the title suggests, I'm interested in a subset of the real line which is neither meagre nor comeagre in any interval. Does anyone have an example? Added. See the comments for some discussion ...
11
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1answer
289 views

Every path has a simple “subpath”

I've been thinking about this for a while, and can't seem to find any way to do it despite the statement itself seeming obvious. The problem is: Let $f:[0,1] \to \mathbb{R}^n$ be a continuous map,...
11
votes
2answers
291 views

Do all manifolds have a densely defined chart?

Let $M$ be a smooth connected manifold. Is it always possible to find a connected dense open subset $U$ of $M$ which is diffeomorphic to an open subset of R$^n$? If we don't require $U$ to be ...
11
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3answers
780 views

variant on Sierpinski carpet: rescue the tablecloth!

I was playing around with Sierpinski carpets (see pretty GPU-produced picture here), and came up with a variation that I have been unable to find mentioned elsewhere. I'm wondering if anyone can tell ...
11
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1answer
287 views

Product of spaces is a manifold with boundary. What can be said about the spaces themselves?

Suppose I have two topological spaces $X,Y$ and I know that $X\times Y$ is homeomorphic to a manifold with boundary. Can I conclude that $X$ and $Y$ are manifolds (maybe with boundary)? If not, ...
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3answers
744 views

What's wrong with this definition of continuity?

Consider this definitions: A function $f:X \to Y$ is continuous at $x\in X$ iff for any open neighborhood $V_{f(x)}$ of $f(x)$ there is an open neighborhood $U_{x}$ of $x$ that gets mapped by $f$ ...
10
votes
3answers
6k views

Union of closure of sets is the closure of the union: true for finite, false for infinite unions

Let $A_i$ be a subset of a metric space for each $i\in \mathbb{N}$. Let $B_n := \bigcup_{i=1}^n A_i$. Prove (for any) $n \in \mathbb{N}$ that $\overline{B_n} = \bigcup_{i=1}^n \overline{A_i}$. ...
10
votes
1answer
295 views

In the Sorgenfrey plane, is an open disc homeomorphic to an open square?

In the sorgenfrey plane $\mathbb{R}_l^2$, the subspace $$X=\{(x,y):x^2+y^2\leq 1\}$$ is not homeomorphic to the subspace $$Y=\{(x,y):|x|\leq 1,|y|\leq 1\},$$ because there is only one isolated point ...
10
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1answer
141 views

Does a map between topologies determine a map between sets?

Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be Hausdorff spaces. Consider a function \begin{equation*} \phi:\mathcal{B}\rightarrow \mathcal{A} \end{equation*} which preserves inclusion, arbitrary ...
10
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3answers
2k views

Is a discrete set inside a compact space necessarily finite?

Is it true that if $A$ is discrete as a subspace of $X$, and $X \;$ is compact, then $A$ is finite? If this doesn't hold, then does it hold for $X\;$ manifold?
10
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2answers
372 views

No Smooth Onto Map from Circle to Torus

My professor was lecturing today and he made this statement which I was unable to verify. (I worded it nicer) There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$. When ...
10
votes
1answer
317 views

Orbit space of a free, proper G-action principal bundle

Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such ...