# 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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### Restriction of quotient map to open subset

Recently I encountered an alleged fact about restrictions of quotient maps, and I tried proving it. Arriving, with some help, at a proof sketch. But today I was told that the fact wasn't true, so I ...
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### Does the quotient map need to take open sets to open sets, why? [duplicate]

The quotient map says that open sets in the image must be open in the preimage, but it says nothing about open sets in the domain needing to map to open sets in the image does it? Otherwise is this ...
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### If $U_1, U_2,…$ is an infinite collection of open sets , then their intersection is open ? True or false?

If $U_1, U_2,.....$ is a infinite collection of open sets , then their intersection is open ? True or false ? I proved that , If $U_1, U_2,......,U_n$ is a finite collection of open sets , then ...
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### Baire's Theorem and Irrationals

I am asked to show that the irrational numbers are not a countable union of closed subsets of $\mathbb{R}$ given that if a complete metric space is the countable union of of closed subsets then at ...
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### Example of homeomorphism of $S^1$ which behaves badly relatively to the Lebesgue measure

Could anyone give an example of a homeomorphism of $S^1$ which sends an open set of full Lebesgue measure on an open set which has not full measure ?
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### prove that intersection of the family of the set is connected

I got no clue to solve this problem, because I can't find the connection between the compactness and the connectedness for the set family. Can anyone help me to solve this? I really appreciate. $X$...
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### Find all interior, boundary, accumulation and isolated points of this set.

Let $S=\{ a\in\mathbb R\mid a \text{ is the root of a polynomial with integer coefficients}\}$. I'm looking to find all of the interior, boundary and accumulation point of $S$. I got that the ...
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### Showing the Hausdorff metric inherits completeness

Let $(X,d)$ be a metric space and let $K(X)$ denote the set of all compact subsets of $X$. Then $(K(X), d_H)$ is a metric space, where $d_H$ is the Hausdorff metric. How can I show that if $X$ if ...
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### Show that there is a invertible continuous function $h: \mathbb{Q} → \mathbb{Q}$ such that $h(−1) = 0$, $h(0) = 1$, $h(1) = −1$.

Show that there is a invertible continuous function $h: \mathbb{Q} → \mathbb{Q}$ such that $h(−1) = 0, h(0) = 1, h(1) = −1$. My attempt so far has been to try to split the rationals in [0,1] into ...
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### Is this metric space complete?

Let $a, b \in \mathbb{R}$ such that $a < b$, and let $X$ be the metric space of all the (real- or complex-valued) functions defined and continuous on the closed interval $[a,b]$ with the metric $d$ ...