# 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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### Morphisms between $\Delta$-complexes?

I recall the definitions I am using. A $\Delta$-complex is a topological space $X$ together with a partition of $X$ into cells and, for every $n$-cell $A$ of the decomposition, a continuous map ...
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### The “removing a point trick”. [closed]

If $X$ and $Y$ are homeomorphic, prove that for any $x\in X$ there exists $y\in Y$ such that $X-\{ x\}$ is homeomorphic to $Y-\{ y\}$. Also explain that, "there exists $y\in Y$" cannot be replaced ...
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### Is the Cartesian square of the set of irrational numbers path connected?

Let $X=\mathbb{R}\setminus \mathbb{Q}$. Is $X\times X$ path-connected? I don't know where to start I think we need some number theory knowledge.
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### How to determine the number of edges and vertices in a plane model

I'm trying to understand the answer posted here: Plane models from the "word" "To determine the Euler characteristic, first compute the vertex cycles and then count how many cycles ...
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### Sum of open and closed sets

Let $A,B$ subsets of a normed space $(X,\|\cdot\|)$ and $A+B=\{a+b\mid a\in A,\, b\in B\}$ I need help with the next proofs, I can't figure how to begin the proofs: (a) If $A,B$ open then $A+B$ open ...
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### finding noninterval sets that are either closed and unbounded or bounded and open

I need to find an example of a closed set $E$ that is not an interval and not compact and a bounded set $F$ that is not an interval and not compact. I'm really confused how to find this without using ...
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### Showing that a metric space on $[0,1]^{\mathbb{N}}$ is compact with one distance function but not another

NOTE: I have no notion of product spaces, so, a proof using the more basic principals would be fantastic. If it helps, I have that a space is compact iff it is sequentially compact iff it is complete ...
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### Measure of the irrational numbers?

I have read that the measure of the irrational numbers on an interval $[a,b] = b-a$. This both makes sense and doesn't make sense to me. If you consider that the union of the irrationals with the ...
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### Find all closed subsets of a given X

Let $X$ = {1, 2, 3, 4, 5} and $\tau$ = {$\emptyset$, {1}, {1,2}, {1,2,5}, {1,3,4}, {1,2,3,4}, $X$}. Show that $\tau$ is a topology and find all closed subsets of $X$. I showed that $\tau$ is a ...
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### What's the general idea behind (rigorously?) proving that a metric space is a manifold?

Perhaps this is a broad question, but I opened Spivak's Differential Geometry and on the first page, it defines a manifold as such: A manifold is supposed to be "locally" like one of these ...
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### when the intersection of the closure of a decreasing sequence of sets has empty interior?

Hi everyone: Suppose that $(A_{n})_{n}$ is a decreasing sequence of subsets of a ball in $\mathbb{R}^{N}$ $(N\geq2)$. If the intersection of all the $A_{n}$'s is empty, can the intersection of their ...
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### Finiteness of set by given topology

Let $X$ be a set. 1) The property of compactness on $X$. 2) If in $X$ we have the discrete topology together with 1), this implies that $X$ is finite. My question: If there exists another ...
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### Is the union of an increasing sequence of topological copies of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?

Let $M$ be an $n$-dimensional topological manifold, and let $(U_k)_{k \in \mathbb{N}}$ be an increasing sequence of open sets $U_k \subset M$ such that for each $k \in \mathbb{N}$, $U_k$ is ...
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### If a power series has positive radius of convergence and is non-constant within radius of convergence , then is all the zeroes of the series isolated?

Let $f(x)=\sum_{n=0}^\infty a_n x^n$ be a real power series with positive radius of convergence $R$ (including $R=+\infty$) , then we know that $f$ is continuous in $(-R,R)$ , so the zero set of $f$ ...
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### Prove that Urysohn spaces are completely Hausdorff

Theorem: Urysohn spaces are $T_{2\frac{1}{2}}$. My attempt: Let $(X, \tau)$ be an Urysohn space. Let $u, v$ be distinct points in $X$. Let $f$ be an Urysohn function for $\{u\}$ and $\{v\}$. ...
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### How to find the preimage of a set

I'm trying to prove continuity of two different functions $f\colon A\to B$. I know that to find continuity, the inverse of all open subsets in $B$ must be open in $A$. I also know that to find the ...
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### Proof on Connectedness

it holds that $X=\prod X_i$ is connected if all the $X_i$ are for an arbitrary index set. Could you check the following proof? I thinks it's wrong, but I can't find any error: Let $X_i$ be connected ...
### $M$ not orientable implying results about $H_{n-1}(M, \mathbb{Z})$, $H_n(M, \mathbb{Z}_q)$?
Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. How do I see that if $M$ is not orientable, then the torsion subgroup of $H_{n-1}(M, \mathbb{Z})$ is cyclic of order $2$ ...