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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6
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1answer
268 views

Nets - preoredered sets or posets?

Question: Usually net is defined as a function from a directed preordered set to a topological space. What would we lose or gain if we worked with partially ordered directed sets only? Background ...
6
votes
2answers
528 views

'Equivalent' Exhaustion by compact sets

Given an open set $U \subset \mathbb R ^n $, there exists an exhaustion by compact sets, i.e. a sequence of compact sets $K_i$, s.t. $\cup _{i=0}^{\infty} K_i = U$ and $\forall i \in \mathbb N : K_i ...
5
votes
1answer
84 views

Example of non-homeomorphic compact spaces $K_1$ and $K_2$ such that $K_1\oplus K_1$ is homeomorphic to $K_2\oplus K_2$

Once I heard that there exists two compact spaces $K_1$ and $K_2$ which are non-homeomorphic, but with $K_1\oplus K_1$ homeomorphic to $K_2\oplus K_2$ (where $\oplus$ denotes the topological sum). Is ...
5
votes
0answers
224 views

How many compatible group structures does a topological space admit?

Suppose I have a topological space $(G,\tau)$ and am interested in whether there exists a topological group $(G,*,\tau)$. In other words, can we assign a binary operation $*$ to $(G,\tau)$ which ...
5
votes
2answers
313 views

When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
5
votes
1answer
291 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
4answers
12k views

Prove that the intersection of a finite number of open sets is open.

More specifically, let $O_1, . . . , O_n$ be a finite collection of open subsets of the continuum, $C$. Then the intersection $O_1 ∩ · · · ∩ O_n$ is open as well. I think it is possible to do it ...
5
votes
1answer
162 views

Topology from interiors of closed sets?

Assume you've got an arbitrary topological space $X$. Now let $I$ be the set of the interiors of all closed subsets of $X$. And now assume you give me $I$, but don't tell me what $X$ is. Can I ...
5
votes
4answers
7k views

If A is a subset of B, then the closure of A is contained in the closure of B.

I'm trying to prove something here which isn't necessarily hard, but I believe it to be somewhat tricky. I've looked online for the proofs, but some of them don't seem 'strong' enough for me or that ...
5
votes
2answers
579 views

Analytic Applications of Stone-Čech compactification

Following Bredon's Topology and Geometry, we let $\mathcal{F}$ be the set of all continuous maps $f:X \to [0,1]$ on a completely regular space $X$, define $X \xrightarrow{\Phi} [0,1]^{\mathcal{F}}$ by ...
4
votes
1answer
99 views

Complement of a point of a Compact Connected Hausdorff Space has no compact maximal connected subspace

This question is a slight modified version of Compact Connected Hausdorff Space has no compact component in the complement of a point Let $X$ be a Hausdorff Compact Connected Space. Prove that ...
4
votes
2answers
103 views

Algebraic condition for a twist in 2D ball or a hypersphere?

We define twisted ball here as a ball where only one point (the twist) separates the ball sides. Simple implicit presentation for the 2D Ball is $x^2+y^2=r^2$. I am trying to find a general condition ...
4
votes
5answers
177 views

Nets and Convergence: Why directed indices?

Please do read carefully (I know Nets-Topology-Filters and their interrelations!!!) 1.) Why do we require nets to be indexed by directed sets (apart from it simply works compared to filters and ...
4
votes
3answers
196 views

Construct a complete metric on $(0,1)$

Can anyone construct a complete metric on $(0,1)$ which induces the usual subspace topology on $(0,1)$ ?
4
votes
1answer
136 views

Etale spaces of a presfeaf and the associated sheaf

Given a presheaf $\mathcal{F}$on a topological space $X$, one can construct the etale space $\pi_1 : Y_1\to X$. Let us now look at the associated sheaf $\mathcal{F}^+$ as a presheaf and construct the ...
4
votes
2answers
540 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
votes
2answers
286 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
votes
1answer
123 views

What is the topology the author used which make $T$ is metrizable?

The paper is Mizokami : On characterizations of spaces with $G_\delta$-diagonals See its Theorem 1, also you can see the picture . http://picpaste.com/a-eaiF4d3t.bmp. Theorem 1: A space $X$ has a ...
4
votes
1answer
246 views

prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.

A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ R_{n}$ and $V ⊂ R_{m}$ are homeomorphic, then n = m. prove that the sphere with a ...
4
votes
1answer
288 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
votes
1answer
100 views

A question on star countable space

Here is a proposition from the paper of L.P. Aiken Star-covering properties. I cannot understand why (the last line) the space is star countable. Could somebody help me? Thanks ahead. The picture is ...
4
votes
2answers
363 views

How to prove this result about connectedness?

Let $X$ and $Y$ be connected, and let $Y \subseteq X$. If $A$ and $B$ are two non-empty, disjoint open sets (open in the subspace $X-Y$) whose union is $X-Y$, or in other words if $A$ and $B$ form a ...
4
votes
3answers
932 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 ...
4
votes
3answers
147 views

Describe a necessary and sufficient condition for the spaces $\mathbb R\setminus A$ and $\mathbb R\setminus B$ to be homeomorphic.

Let A and B be two finite subsets of $\mathbb R$. Describe a necessary and sufficient condition for the spaces $\mathbb R\setminus A$ and $\mathbb R\setminus B$ to be homeomorphic. I think $|A|=|B|$. ...
4
votes
1answer
1k 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?
3
votes
2answers
71 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 & = ...
3
votes
2answers
82 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
3answers
188 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
votes
1answer
167 views

“isometric invariant” vs “isometric” what do these term mean?

I am now hopelessly confused: There is Hilberts Theorem https://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29 . that implies that there are no isometric embeddings of the ...
3
votes
2answers
85 views

A topological space which is Frechet but not Strictly-Frechet.

Let $X$ be a topological space and $q \in X$. $X$ is strictly Frechet at $q$, if, for all $A_n \subset X, q \in \bigcap_{n \in \omega} \overline {A_n}$ implies the existence of a sequence $q_n \in ...
3
votes
1answer
776 views

Definition of a nowhere dense set

I'm currently studying metric spaces through Gamelin and Greene's Introduction to Topology. While studying about completeness I got stuck with this concept of nowhere dense subset. The book defines a ...
3
votes
1answer
606 views

Open sets in product topology

For any two topological spaces $X$ and $Y$, consider $X \times Y$. Is it always true that open sets in $X \times Y$ are of the forms $U \times V$ where $U$ is open in $X$ and $V$ is open in $Y$? I ...
3
votes
2answers
137 views

Does the given $h$ exhibit the homeomorphism between $\mathbb{R}^{\omega}$ and itself with box topologies?

(Munkres, p. 118, Problem 8) Give $h: \mathbb{R}^{\omega} \to \mathbb{R}^{\omega}$ $$ h(x_1,x_2,\dots) = (ax_1 +b,ax_2+b,\dots) $$ where $a,b\in \mathbb{R}$. Does the given $h$ exhibit the ...
3
votes
2answers
255 views

Is there a first countable, 0-dimensonal, locally compact, lindelöf, non-compact space?

Is there a first countable, 0-dimensonal, locally compact, lindelöf, non-compact space in which all non-empty open sets have $\pi$-weight $\mathfrak c$? It also can be seen here. Thanks for your ...
3
votes
0answers
84 views

A connected sum and wild cells

Can we find such a connected sum of two spheres (in any dimension) that is not homeomorphic to the sphere? $\def\R{\mathbb R}$ It seems that there should be examples like that, because there are lots ...
3
votes
3answers
342 views

On the existence of a continuous bijection from a quotient space to the unit sphere $S^2$

There is a question from an old topology prelim that is somewhat giving me a hard time. Consider the cylinder $X= S^1 \times [-1,1]$. Now we define an equivalence relation $\sim$ as follows: For ...
3
votes
2answers
221 views

The Stone-Čech compactification of a space by the maximal ideals of the ring of bounded continuous functions from the space to $\mathbb{R}$

There is a claim that for any completely regular space, the maximal ideals of the ring of bounded continuous functions from $X$ to $\mathbb{R}$ forms the Stone-Čech compactification of $X$. How is the ...
3
votes
1answer
71 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 ...
3
votes
1answer
126 views

What is Baire's zero-dimensional metric space?

I'm not familar with metrizable spaces. I met a notation: Baire's zero-dimensional metric space. Could somebody explain it for me? Thanks ahead:)
3
votes
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 ...
3
votes
1answer
938 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 ...
3
votes
2answers
288 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
3
votes
1answer
477 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
votes
1answer
618 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, $B_0 = \bigcap_{\epsilon>0} B_{\epsilon}$. ...
3
votes
2answers
204 views

If a connected open set is evenly covered, then its preimage is uniquely partitioned into slices

This is from Topology by Munkres: Let $p:E \to B$ be a covering map. Suppose $U$ is a open set of $B$ that is evenly covered by $p$. Show that if $U$ is connected, then the partition of ...
2
votes
2answers
88 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
2answers
655 views

How to complete this proof to show that the metric $d'(x,y) = d(x,y) / (1 + d(x,y))$ gives the same topology as $d(x,y)$ gives?

This is an exercise problem from Munkres's Topology (Exercise 11 of Section 20 "The Metric Topology", 2nd edition). Exercise 11: Show that if $d$ is a metric for $X$, then $$d'(x,y) = d(x,y) / (1 ...
2
votes
6answers
269 views

Prove that $[a,b]$ is connected space.

Prove that $[a,b]$ is connected space. I know that $\mathbb{R}$ with euclidean metric is connected space. I would like find surjective function $f: \mathbb{R} \rightarrow [a,b]$. Because $\mathbb{R}$ ...
2
votes
2answers
97 views

On continuously uniquely geodesic space

This question was inspired by this comment of @68316. Definition : A continuously uniquely geodesic space is a uniquely geodesic space whose geodesics vary continuously with endpoints. Question ...
2
votes
1answer
903 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) ...