# Tagged Questions

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### Prove that $\mathbb{N}$ is not metrizable where $U$ is open if it is $U=\mathbb{N}$, $U = \emptyset$, or $\mathbb{N}$ \ $U$ is a finite subset.

$\mathbb{N}$ is the set of natural numbers. Any set $U$ is open if it is $U=\mathbb{N}$, $U = \emptyset$, or $\mathbb{N}$ \ $U$ is a finite subset. This defines a topology on $\mathbb{N}$. Prove ...
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### Compact subset of a closed subspace: compact in the whole space?

Imagine that you have a topological space $(X,\tau_{X})$ and a closed subset $Y$. Say that within $Y$ we have a subset $K$ that is compact in the subspace topology $\tau_Y$. Is $K$ compact in ...
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### Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
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### understanding topological argument in rado-kneser theorem

Rado-kneser choquet theorem states that Poisson integral of a homeomorphism of unit circle is a homeomorphism. It's proof goes like proving it local homeomorphism by proving non vanishing of jacobian ...
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### what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
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### Can we take the infimum over a variable set?

Suppose that we have a family of functions $\lbrace f_{\alpha}(x)\rbrace_{\alpha}$ define on an open set of $\mathbb{R}^{m}$, and $\alpha$ runs over a set $\Gamma$. Assume that the family is ...
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### Order-preserving embeddings

(Follow-up to Existence of a utility function on the reals.) Say we have a totally ordered set $X$ which has a countable, dense subset $C$. I believe we can find an $f:C\to\mathbb R$ which is ...
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### Every Bounded set contained in a Compact set

In a general metric space, is every bounded set contained in a compact set?
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### Is every separately continuous function on $R^2$ continuous?

My friend asked me this question a few days ago. I felt it's not right but couldn't find a single counterexample. Any comment is appreciated.
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### Extending $C \cong C \times C$ to the Space-Filling Curve $[0,1] \cong [0,1]^2$

Setting: Let $C$ denote the Cantor Set. I have already shown that $C \cong \prod_{i=1}^\infty \{0,2\}_i$, and make use of this fact below. Goal: Show that $C \cong (C \times C)$, and then use this ...
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### A question on accumulation points

Suppose $S$ is a closed, countable, subset of $\mathbb{C}$. For any $n$, define $S_n:=S_{n-1}\backslash B_{n-1}$, where $B_{n-1}$ is the set of isolated points of $S_{n-1}$ (as usual, $S_0:=S$). Is it ...
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### Showing a Space-Filling Curve is Continuous

Setting: Let $F':[0,1] \rightarrow [0,1]$ be the Cantor Function. Goal: Show that there exists a well-defined, surjective, continuous function from $[0,1]$ to $[0,1]^2$ (i.e., a space-filling curve). ...