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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0
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1answer
89 views

Finite topological space

At http://math.stackexchange.com/questions/1528995/finite-topological-space the user asked "Let $X$ be a Hausdorff topological space such that every closed subset has finitely many connected component....
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1answer
46 views

Prove that a function from a metric space into [0,1] is continuous and real valued.

$(M,d)$ is a metric space and $X$ is a subset of $M$. define $d(x,X) = \inf \{d(x,y)| y \epsilon X\}$ A and B are closed subsets of M such that $A \cap B = \phi$ a. Prove that $$g(x) = \frac{d(x,A)}{...
2
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0answers
33 views

Definition of locally connected metric space

I have this definition of locally connected metric space: "A metric space $(X,d)$ is called locally connected if for all $x\in X$ and for all $U\subset X$, $U$ neighbourhood of $x$, exists a connected ...
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votes
0answers
20 views

Advice to get started - Upper and lower hemicontinuity

Consider the correspondence $ f \colon \mathbb{R} \to \mathbb{R}$ defined by: $f(a) = \{x \in \mathbb{R}:x^2 + x \leq a^2\}$ for all $a \in \mathbb{R}$. Find the points where the correspondence is ...
0
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1answer
29 views

Two sets S and T are disjoint and compact in a normed vector space. Define $f(S,T)=\inf\{||s-t||:s \in S, t \in T\}$. [duplicate]

Two sets S and T are disjoint and compact in a normed vector space. Define $f(S,T)=\inf\{||s-t||:s \in S, t \in T\}$. Are there elements $s \in S$ and $t \in T$ s.t. $f(S,T)=||s-t||$?
1
vote
1answer
98 views

Find the derived set of $\{\frac{1}{n} + \frac{1}{m}: m,n \in \mathbb{N}\}$ and prove it is such.

It is easy to see the derived set is $A' = \{\frac{1}{n}: n \in \mathbb{N}\}\bigcup\{0\}$. To prove these are the only elements of the derived set we need to show that the shape of the derived set can ...
5
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1answer
74 views

Chain map induces map of chain complexes and induced product, intuition behind isomorphism preserving products?

I know there is an isomorphism$$H^*(K(\pi, 1), A) \cong \text{Ext}_{\mathbb{Z}[\pi]}^*(\mathbb{Z}, A).$$When $A$ is a commutative ring, the $\text{Ext}$ groups have algebraically defined products, ...
0
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1answer
37 views

$\overline{\operatorname{int}(A)} \subseteq A$ for all closed sets $A \subseteq \mathbb R$

Is $\overline{\operatorname{int}(A)}$ (the closure of the interior of $A$) contained in $A$ for all closed sets $A \subseteq \mathbb R$ true? How can I prove it?
5
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2answers
45 views

$X$ space satisfying hypotheses used to construct a universal cover, $A$ abelian, $C^*(X, A) \cong \text{Hom}_{\mathbb{Z}[\pi]}(C_*(\tilde{X}), A)$?

Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\tilde{X}$ and let $A$ be an abelian group. What is the most elementary way to see that$$C^*(X, A) \cong \text{Hom}...
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1answer
16 views

The set A is in a normed vector space W. $S=\bar A \cap \bar {A^c}$.

The set A is in a normed vector space W. $S=\bar A \cap \bar {A^c}$ (S is the intersection of the closure of A and the closure of $A^c$). Is there a set A in W=R for which S=the set of rational ...
0
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0answers
25 views

Some questions about $[0, 1]^{2^\mathfrak c}$

I was reading an article that made me think about some questions regarding $[0, 1]^\mathfrak c$. I know that there exists a dense subset of $X=[0, 1]^{2^\mathfrak c}$ of cardinality $\mathfrak c$. ...
0
votes
2answers
32 views

Notation for continuous functions

Let's say $(X,\sigma)$ and $(X,\tau)$ are topological spaces, and $f$ is a continuous function from the former to the latter. (That is, the inverse images of elements of $\tau$ are elements of $\sigma$...
2
votes
1answer
40 views

Why is a continuous injective map of a closed ball in $\mathbb{R}^2$ nulhomotopic?

In Munkres' Topology, the proof of theorem 62.3 goes as follows: Let $U$ be an open subset of $\mathbb{R}^2$ and let $f:U\rightarrow S^2$ be continuous and injective. Then let $B$ be any closed ball ...
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vote
2answers
65 views

Showing that 3 dimensional unit sphere is connected

Let $$\{(x,y,z)\in \Bbb R^3, x^2+y^2+z^2=1\}$$ I need to show that this set is connected. I have tough time handling connectedness. Help would be appreciated! Thanks in advance!
1
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2answers
25 views

Limit of sequence using mean value

Let $f(x)=1-\frac{x}{10}$ and define the sequence $(a_n)$ by: \begin{align} & a_0 = 0, \\ & a_{n+1} = f(a_n), \text{ for } n\geq 0. \end{align} I found $x_0$ such that $f(x_0) = x_0$ which ...
0
votes
1answer
29 views

Show $g=3$ is contained in the unit ball of $X=C[0,1]$ in the uniform metric.

How do I show that the function $g=3$ is contained in the unit ball of $X=C[0,1]$ in the uniform metric? I know the uniform metric $$d_u(g,0)=sup |3|$$ where do I go after this?
0
votes
2answers
50 views

Size of the deck transformation group

If $p\colon Y\to X$ is a $k$-fold covering map, and $Y$ is path-connected, what is the size of Deck($p$), the deck transformation group? I was attempting to prove that the answer is $\leq k$, but ...
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votes
2answers
34 views

Connected and disconnected sets

If A and B are connected, then does it imply that their union is connected? For example, if $A=[0,2], B=[4,5]$. Is the union connected? I believe the intersection is not necessarily connected, but how ...
1
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0answers
29 views

Quintic equation and number of lines on the quintic

I heard a talk where the speaker said that the solution to the equation $x_1^5 +x_2^5 +x_3^5 +x_4^5 +x_5^5 = 0$ is a six-dimensional (Calabi-Yau) manifold. Then he went on to define five curves of ...
0
votes
1answer
41 views

Limit of accumulation points

Suppose $S \subset \mathbb{R}$ is a set and, for each $n$, $x_n$ is an accumulation point of $S$. Suppose further that $\lim_{n}x_{n}=x$. Prove that $x$ is also an accumulation point of S. I was ...
2
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0answers
48 views

Filters and their refinements vs nets and their subnets [duplicate]

True or false? a) Let $(x_{\alpha})_{\alpha\in A}$ a net over a space $X$ and $(x_{h(\beta)})_{\beta\in B}$ a subnet, where $h:B\to A$ is monotone and final. Let $\mathcal{F_1}$ be the filter ...
0
votes
1answer
31 views

Finding the uniform and $L^{1}$ metric between two functions.

I know the uniform metric between two functions f and g is defined as $$d_u (f,g)=sup|f(x)-g(x)|$$ and the $L^{1}$ is defined as $$d_1 (f,g)=\sum^{n}_{i-1} |f-g|$$ Say that $$X=C[0,1]$$ Lets say I ...
0
votes
3answers
58 views

cauchy sequence on $\mathbb{R}$

i want to show that $\mathbb{R}$ with the following metric : $d_1(x,y)=|x^3-y^3|$ is complete. I think a good way to show it is to show that a sequence which is Cauchy for $d_1$ will also be Cauchy ...
0
votes
1answer
23 views

Accumulation points and bounded open sets

Suppose $S$ is a bounded open set in the reals and suppose that the least upper bound of $S$ is $7$. Prove that $7$ is not an element of $S$. Use the definition of accumulation point to prove that $7$...
-1
votes
1answer
35 views

Proving that the union of connected sets is connected when consecutive sets have nonempty intersection

Let $\{S_n: n=1,2,\dots\}$ be the collection of connected sets with the property that $S_n\cap S_{n+1} \neq \emptyset$. Prove that $\bigcup_{n=1}^\infty S_n$ are connected. Help would be appreciated!
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1answer
116 views

Set of all unitary matrices - compactness and connectedness.

Let U denote the set of all nxn matrices A with complex entries such that A is unitary. Then U as a topological subspace of $C^{n^2}$ is a) compact but not connected. b) connected but not compact. ...
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2answers
89 views

Proving the cartesian product of $n$ sets is closed.

I'm having trouble with this. Suppose that $U_1, U_2, \dots U_n$ are closed subsets of $\mathbb{R}^n$. I know that each $U_i' \subseteq U_i$ where $U_i'$ is the derived set but this doesn't feel like ...
0
votes
1answer
45 views

Show that any collection of disjoint open subsets of $\mathbb{R}^n$ is countable.

Looking for feedback on my proof. Thanks! Suppose $G = \{A_i\}_1^\infty$ is a collection of pairwise disjoint open subsets of $\mathbb{R}^n$. let $x$ be a point in $G$. Since each $A_i$ is open there ...
2
votes
2answers
54 views

Prove that $ U=\{f\in C[0,1]: f(x)\neq 0, \forall x \in [0,1]\}$ is open and find his connected components

In $(C[0,1],d_\infty)$, consider $U=\{f\in C[0,1]: f(x)\neq 0, \forall x \in [0,1]\}$. Prove that $U$ is open and find his connected components. I know that for proof the first thing, i have to show ...
1
vote
1answer
27 views

showing $Ax=b$ has a unique solution by finding the fixed points of a function

Let $n ∈ \mathbb N$. Consider an $(n×n)$-matrix A with real components and a column vector $b ∈ \mathbb R^n$. They give rise to an affine transformation $T : \mathbb R^n → \mathbb R^n$ with $T(x) = Ax+...
8
votes
4answers
953 views

Is every manifold a metric space?

I'm trying to learn some topology as a hobby, and my understanding is that all manifolds are examples of topological spaces. Similarly, all metric spaces are also examples of topological spaces. I ...
1
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0answers
37 views

Distinctions of different topologies on the sequence space (countable cartesian products of $\mathbb{R}$)

$\newcommand{\b}[1]{\mathbf{#1}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\N}{\mathbb{N}}$ Question I solved this exercise in Munkres.(20.4) But I don't know if I did it righ t or not. I really ...
3
votes
1answer
72 views

Surface groups and subgroups of fundamental groups

The fundamental group of any closed surface is a surface group. Let $S_3$ be the orientable surface of genus 3. Is $\pi_1(S_3)$ isomorphic to an index-3 subgroup of any surface group? We have 1 2-...
0
votes
1answer
84 views

Is there any continuous surjective map from $[0,1) $ to $\mathbb R$?

Is there any continuous surjective map from $[0,1) $ to $\mathbb R$? If it exits then what is the way to construct such a map?
0
votes
1answer
40 views

Open and Closed Set intervals

What is an example of an open set $A_1$ in the reals which contains the interval (1,2) but so that $A_1$ is not itself an interval. Find another $A_1$ except this time $A_1$ is a closed set ...
2
votes
1answer
29 views

Metric topology but not group topology

I am looking for a metric $d$ that defined on a group $X$ but $(X,d)$ fails to be a group topology. For the semimetric, the following construction will do: Consider the additive group ${\bf{Z}}$ with ...
0
votes
1answer
52 views

countable dense subset of R^k

This is a question from Rudin's Principles. Chapter 2, question 22. The question reads: "A metric space is called $separable$ if it contains a countable dense subset. Show that $\mathbb{R}^k$ is ...
1
vote
1answer
74 views

A 2-sphere bounding a 3-ball?

I'm reading Hatcher's "Notes on Basic $3$-Manifold Topology" and he quickly refers to the notion of a $2$-sphere embedded in a $3$-manifold, which bounds or not an embedded $3$-ball, but without ...
4
votes
1answer
59 views

Show that if a trapping region Q is path connected, the basin of Q must also be path connected

Consider a smooth bijection $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$. A trapping region is a bounded subset $Q$ of $\mathbb{R}^2$ with the properties int$Q\neq \emptyset$, $F(Q)\subseteq Q$, and $F(Q)\...
2
votes
1answer
117 views

connected components in box topology

Consider $R^{\omega}$ in the box topology. x & y are in the same component iff $x_i = y_i$ for infinitely many values. I proved the --< implication, but I can't come up with the proof for ...
1
vote
0answers
48 views

Show that there exist Borel sets $B_n$ such that $B=\bigcup B_n$

Let $X$ be a Polish space. Let $B$ be a Borel subset of $X \times X$ with the following property: $$\forall x \in X \ \left|\left\{y : (x,y) \in B\right\}\right| = \aleph_0.$$ Show that there exists ...
0
votes
1answer
39 views

Why are these sets disjoint? (Munkres' Topology)

There's a step in a proof in Munkres' Topology that doesn't make sense to me. Theorem: Let A be a connected subspace in X. If $A \subset B \subset \bar{A}$, then B is also connected. Proof: Let A be ...
3
votes
1answer
35 views

Deciding the intervals are open or closed

Let $D^+=\{(x,+{\infty}): x \in \mathbb R\}\cup\emptyset$. In $(\mathbb R,D^+)$, check whether the following intervals are open, closed, both open and closed, neither open nor closed. $A=(-\infty,2]...
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3answers
84 views

Is the complement of a compact set closed? [closed]

Suppose that $U$ is compact and $V=\Bbb R^n \setminus U$. Now, is V closed or neither open nor closed?
5
votes
2answers
316 views

Example of a topological space which is not first-countable

According to Munkres' Topology: Definition. A space $X$ is said to have a countable basis at $x$ if there is a countable collection $\mathscr B$ of neighborhoods of $x$ such that each neighborhood ...
1
vote
2answers
31 views

All the spaces which are Hausdorff and semi metric spaces, are necesarily metric spaces?

My hypothesis: $(X,d)$ is a semi metric space and a Hausdorff space My thesis: $d$ is metric My try: Well $d$ is semi metric so, to prove that $d$ is metric, I only have to prove that: $d(x,y)=0 \...
0
votes
1answer
25 views

Proof of $\overline{A}=\prod_{i\in I} \overline{A_i}$ (Topological product)

Can someone explain me this proof please: We have $\prod_{i\in I} \overline{A_i}=\cap_{i\in I} F_i$ where $F_i=\prod_{j\in I} B_j$ with $B_i=\overline{A_i}$ and $B_j=E_j$ if $j\neq i$. We ...
2
votes
1answer
63 views

Finite topological space

Let $X$ be a Hausdorff topological space such that every closed subset has finitely many connected component. How can I verify that $X$ is finite?
0
votes
0answers
12 views

Proof: A point is a limit point of S iff a sequence in S converges to it [duplicate]

I had a go at this proof; just wanted to see if it is correct reasoning. Thanks! Question Prove that $x$ is an accumulation point of a set $S$ iff there exists a sequence $(s_{n})$ of points in $S\...
0
votes
1answer
26 views

Proof about equality of sets (increasing sets)

Let $B$ be some subset of the metric space $(X,d)$. Suppose there exists closed subsets $\{ B_{n} \}_{n=1}^{\infty} \subset X$, such that $B = \bigcup\limits_{n=1}^{\infty}B_{n}$. Now let's say I ...