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condition $\mu ^*(B\cup C)=\mu ^*(B)+\mu ^*(C)$ |
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Let $\displaystyle f$ be a continuous real valued function on the metric space $\displaystyle (X,d)$. |
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Cardinality of the collection of all compact metric spaces |
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Continuity Properties |
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Suppose that $x$ is a fixed nonnegative real number such that for all positive real numbers $E$ , $0\leq x\leq E$. Show that $x=0$. |
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Prove $\operatorname{dist}(\overline{A},\overline{B}) = \operatorname{dist}(A, B)$ |
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$x_n \in \mathbb R, \quad x_n \to A \implies \max \{x_n,A-x_n\} \to ?$ |
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Are these subsets of $\mathbb{R}$ homeomorphic? |
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Let $ (x_n) $ be a divergent sequence in a compact subset of $ \mathbb{R}^n $. Prove there are two subsequences that converge to different limits. |
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Alternative proof of Monotone Convergence Theorem |
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Quotient map for usual topology |
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Understanding Complete Metric Spaces and Cauchy Sequences |
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Help understanding $T_1$ & $T_2$ spaces. |
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The space of continuous, bounded functions from a metric space $X$ to $\mathbb R$ |
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Closure of the set $\{\sin(n): n\in\mathbb{N}, n > 0\}$ is the interval of reals $[-1,1]$? |
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Compactness in subsets of $\mathbb{R}^2$ example |
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questions on a continuous, injective, surjective |
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Show that $d_b(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric. |
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How to show that a point is not an interior point? |
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Proof that the interval and the plane is not homeomorphic |
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