# 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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### Explicit construction of an $\epsilon$ net covering

Suppose $X$ is a compact space. In particular $X$ is totally bounded and there exists $x_1,..,x_n$ such that $$X = \bigcup_{i=1}^n U(x_i, \epsilon)$$ where $U$ is the Open Ball centered at $x_i$ ...
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### How to prove that there are only two kinds of 1-dim manifolds without boundary

I just know a conclusion that all 1-dim manifolds without boundary is homomorphism to $S^1$ or $\mathbb{R}$ , but I don't know how to prove it . Why is so ?
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### Lebesgue measure, do we have $m(x + A) = m(A)$, $m(cA) = |c|m(A)$? [closed]

Suppose $m$ is Lebesgue measure. Define $x + A = \{x + y : y \in A\}$ and $cA = \{cy : y \in A\}$ for $x \in \mathbb{R}$ and $c$ a real number. Let $A$ be a Lebesgue measurable set. I have two ...
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### Lebesgue-Stieltjes measure corresponding to a right continuous increasing function, $m(\{x\}) = \alpha(x) - \alpha(x-)$ for each $x$

Let $m$ be Lebesgue-Stieltjes measure corresponding to a right continuous increasing function $\alpha$. How do I see that for each $x$, we have$$m(\{x\}) = \alpha(x) - \alpha(x-)?$$
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### Orthogonal lines on Mercator projection?

I am currently struggling with the following task: We have two pairs of latitude/longitude which determine a small line segment It is needed to get two pairs of latitude/longitude for a small line ...
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### How do I prove that for any finite subsets A and B exists one set R?

How do I prove that for any finite subsets A and B exists one set R $\left | A\cup B \right |=\left | A \right |+\left | B \right | -\left | A\cap B \right |$ Deduce from this an adequate formula ...
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### bounded components of the intersection of two planar domains

It seems to be intuitively clear that if U is a domain in the plane having a bounded complementary component C, then C is also a complementary component of the intersection of U with an open disk D ...
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### Over sequential spaces and $B(H)$

We say that a topological space $X$ is sequential if the following holds : If $U$ is sequentially open then $U$ is open. By sequentially open we mean that $x \in U$ and $x_n \to x$ implies that $x_n$ ...
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### Showing that $\sigma$-algebra is uncountable [duplicate]

Suppose $\mathcal{A}$ is a $\sigma$-algebra with the property that whenever $A \in \mathcal{A}$ is nonempty, there exist $B$, $C \in \mathcal{A}$ with $B \cap C = \emptyset$, $B \cup C = A$, and ...
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### Is $\bigcup_{i = 1}^\infty \mathcal{A}_i$ necessarily a $\sigma$-algebra? [duplicate]

Suppose $\mathcal{A}_1 \subset \mathcal{A}_2 \subset \ldots$ are $\sigma$-algebras consisting of subsets of a set $X$. Is $\bigcup_{i = 1}^\infty \mathcal{A}_i$ necessarily a $\sigma$-algebra? If not, ...
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### If $p: E \rightarrow X$ is a covering map with $E$ connected and $|p^{-1}(x_{0})|=k$ for some $x_{o}$ then $|p^{-1}(x)|=k$ for all $x \in E$.

Prove that if $p:E \rightarrow X$ is a covering map with $E$ connected and $p^{-1}(x_{0})$ has $k$ elements for some $x_{0} \in X$, then $p^{-1}(x)$ has $k$ elements for every $x \in X$. Is my proof ...
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### boundary of starred compact set in the plane.

Is the boundary of a starred compact C set in the $\mathbb R^2$ connected? Let also suppose it to be the closure of an open set.
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### Indicator function for limsup, liminf [duplicate]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...
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### An exact sequence of compact topological groups.

Let $A, B, C$ be abelian topological groups such that we have the following exact sequence : $$0\to A \to B \to C \to 0.$$ Assume also that A, C are compact and all the maps are open. Then it's it ...
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### Proving some specific space to be normal [duplicate]

Let $f:X\rightarrow Y$ be a closed continuous surjection. Assume that $X$ is normal. Prove that $Y$ is normal. $X$ is normal,then every one-point set in $X$ is closed,so is one-point set in $Y$ ...
Let $(X,d)$ be a metric space. It is complete if every Cauchy sequence for $d$ on $X$ is convergent. I've heard an alternative definition of completeness for $(X,d)$: it is complete iff the ...