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The maps $F_K$ are certainly upper semicontinuous, because, by the regularity property of the Lebesgue measure, for every $x\in M$ and for $\epsilon>$ there exists $\delta>0$ such that the uniform $\d …

Let's look more closely at the inclusions you wrote. The first, $C\subset L^1$, is dense w.r.to the $L^1$-norm (for this, it is sufficient to note that the $\|\cdot\|_1$-closure of $C$ contains the c …

The assumption on $(f_i)_i\subset U$ can be stated equivalently: For all $h\in C([0,1]^2)$, the sequence $\int f_ih $ is Cauchy on $\mathbb R$, or also, equivalently: For all $h\in C([0,1]^2)$ it is c …

To rephrase the question in simple terms. Let's use two different intervals $I:=[a,b]$ and $J:=[c,d]$ for more clarity. Let a sequence $(f_n)_{n\in\mathbb N}$ of integrable functions on the rectangle …

Here is, as an example, a sort of "mosquito coil" shaped curve $X\sim\mathbb R$ in $\mathbb R^3$, endowed with the induced Euclidean norm. Let $\Gamma \subset \mathbb R^2$ a pressed logarithmic spira …

A fancy decomposition of the closed unit disk $X$ of $ \mathbb{C}$ into two dense contractible sets. Let $D_1$, $D_2$ be a $2$-partition of the interval $[0,1]$ into dense sets. Consider the disjo …

Since the conversation seems to focus on the density issue in Martin M. W.'s proof: yes, density is really as obvious as the openness. If $h^s(t)$ is a free homotopy in $\mathbb R^3$ between loops $h …

If you like to see the Tietze theorem from a Functional Analysis viewpoint, another possibility is by means of the following basic lemma (extracted from the original Urysohn proof), stating that a cer …

Let $B_r(y)\subset\mathbb C$ denote the open disk of radius $r$ and center $y\in\mathbb C$, and $B_r:=B_r(0)$. Let $h:B_1\to U$ a homeomorphism (e.g. a Riemann mapping) . Then $\Gamma_r:=h(\partial …

On the matter of what definition is preferable. Let me add that the same question may be asked for the countably version, that has $\omega$-accumulation point in place of limit of a subsequence: $A$ …

A bump function on a Banach space $X$ is a non-zero function with bounded support. Existence of a bump function of a given smooth regularity has, of course, strong immediate consequences --by translat …

Here is a counterexample for $K:=[0,1]$. There exists a Borel set $B\subset K$ that meets every non-empty open set $A\subset K$ in a subset of it of positive, not full Lebesgue measure: $0<|A\cap B|<| …

Let $I_0$ be the interval where $f$ is a horizontal motion, and let $I_1$ the interval where it is a circular motion. So up to reparametrisation (and adopting complex notation) $I_0=[-3,0]$ and $I_1= …

Here's a variant of Fernando Muro's construction. Let $X$ be the closure in $\mathbb R^2$of the graph of the function $g(x):=x\sin\big(1/\sin(1/x)\big)$ defined on $[0,+\infty)\setminus\{1/n\pi:n\in \ …

If $c\notin\mathfrak{A}$ in general it is not true: e.g.: consider the case where $X$ is the real line , $\mathfrak{A}$ is the algebra of the polynomials, and $c$ is the function $-e^x$. Then $\mathfr …