# Tagged Questions

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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### 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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### 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 ...
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### 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 $\mathfrak c$. ...
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### 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 $\sigma$...
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### 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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### 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!
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### 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 ...
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### 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?
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $7$...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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?