# Tagged Questions

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### Rudin Theorem 2.7

Theorem 2.7 in Rudin's Real and Complex analysis Theorem Suppose $U$ is open in a locally compact Hausdorff space X, $K \subset U$, and $K$ is compact. Then there is an open set $V$ with compact ...
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### Sets which are open “modulo a nullset”

A set $A$ is said to have property of Baire there exists an open set $U$ such that $A\triangle U$ is meager. So this says that symmetric difference of $A$ and some open set is small (in the sense of ...
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### Hausdorff dimension mathces Box-counting dimension

I need to compute the Hausdorff dimension of certain sets using a computer and, to date, my approach has been to use a Box-counting algorithm, for I once read that the Hausdorff dimension of an ...
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### Proof strategy - Borel $\sigma-$fields

How does one go about proving the following: Every open set $A$ in the topological space $(\mathbb{R}^d,\|\cdot\|)$ (with the norm topology) is the union of all the open balls $B_\epsilon(q)$ whose ...
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### Continuous modification of functions with a given property

Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ with the following property: For all reals $x$, $\displaystyle\lim_{y \to x} f(y)$ exists. (In particular, note that its possible that ...
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### I have a question about Lebesgue measure

Suppose $A$ is Lebesque measurable, show that for each $\epsilon>0$, there exists an open set, $O$, such that $A\subset O$, and $\lambda(O-A)<\epsilon$. (note: $\lambda$ is the Lebesgue measure ...
I gave a short presentation on Baire spaces, and one of the cute results of the theory I showed is that a Vitali set cannot be nowhere dense. This led me to think that a subset of the reals $A$ which ...