# Tagged Questions

Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

27 views

### $\overline{X\cap Y}\subset \overline{X}\cap\overline{Y}$ for real numbers, case when $\overline{X\cap Y}\neq \overline{X}\cap\overline{Y}$

My proof for this is similar to this one, but I can't find an example such that $\overline{X\cap Y}\neq \overline{X}\cap\overline{Y}$ for the real numbers.
27 views

### Together with the algebra of cardinal numbers, is there analysis of cardinal numbers? [closed]

Let $C$ be the collection of all cardinal numbers. Is there any norm, inner-product, metric (other than discrete metric), topology(other than discrete, co-finite topology) on $C$, which is very useful?...
28 views

### for $X\subset \mathbb{R}$, $\mathbb{R} = int X\cup Int(\mathbb{R}-X)\cup \partial X$

I need to prove: for $X\subset \mathbb{R}$, $$\mathbb{R} = int X\cup Int(\mathbb{R}-X)\cup \partial X$$ The problem is that all the proofs I've found are for metric spaces, not $\mathbb{R}$ itself, ...
45 views

42 views

### Proving that the subset is the set itself

I am trying to prove the following property, which seems fairly intuitive, at least in $\mathbb{R}^n$. Let $(X,d)$ be a compact metric space where $Y \subseteq X$ arbitrary. Prove that if there ...
24 views

### Example of a locally compact metric space which is $\sigma$-compact but not proper

Let $(X,d)$ be a locally compact metric space. Then it is known that $X$ is separable if and only if it is $\sigma$-compact (i.e. it can be written as a countable union of compact sets). Moreover, ...
2k views

### May a 'ball' that has been 'cut off' still be called a 'ball'?

Consider the metric subspace $[0,1] \subseteq \mathbb{R}$ with the metric defined in the usual sense, and the ball $B(0,1)$, defined to be the ball centred at $x=0$ with radius $1$. Now since only ...
35 views

### If $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$ then is $(X,d_1)$ homeomorphic to $(X,d_2)$?

Suppose that $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$. Is it always the case that $(X,d_1)$ is homeomorphic to $(X,d_2)$? I have been trying to find a counter example, but ...
27 views

53 views

### Euclidean distance between $x\in\mathbb{R}$ and $\{x\in\mathbb{R} \mid f(x)=0\}$ [closed]

Is there a generic formula to calculate the distance between an arbitrary real number $x\in\mathbb{R}$ and $$\{x\in {\mathbb{R}}\mid f(x)=0\}$$ where we have little information about $f$? In fact, my ...
40 views

### Show that $\bar{A}=\{x \in M | d(x,a)=0\}$

Let $(M,d)$ be a metric space. Let $A$ be an arbitary subset of $M$ and let $x$ be an arbitary point. Define $d(x,A)=\inf \{d(x,y)\mid y \in A\}$. Show that $\bar{A}= \{x \in M \mid d(x,A)=0\}$ How ...
54 views

48 views

18 views

### $K$ is compact and $x\in X$ but $x\notin K$. Show $\exists G_1,G_2$ open in $(X,d)$ s.t. $x\in G_1$ and $K\subseteq G_2$

Suppose $K$ is a compact subset of a metric space $(X,d)$ and $x \in X$ but $x\notin K$. Show that there exist two disjoint open sets of $G_1$ and $G_2$ of $X$ such that $x\in G_1$ and $K\subseteq G_2$...
29 views

### Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$

I have the following exercise: Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$ I don't know what $b$ is meant to be, there's a typo in this exercise. I ...
91 views

### Proof that a discrete space (with more than 1 element) is not connected

I'm reading this proof that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton ${x}$ and its complementar....
23 views

### $A$ is an open subset of $M$ $\iff$ ($x_n\to a\implies x_n\in A$ for large $n$)

My definition of an open subset $A$ of $M$ is the one that for every $x\in A$, there is an open ball contained in $A$. Now, suppose that $x_n\to a$. By definition, $\forall \epsilon>0$ there exists ...
31 views

25 views

### Questions about proof of $\lim x_n = a, \lim y_n = b\implies \lim x_n+y_n = a+b$ in a normed vector space

I need to prove that, in a normed vector space $E$, we have: $$\lim x_n = a, \lim y_n = b\implies \lim (x_n+y_n) = a+b$$ and: \lim\lambda_n = \lambda, \lim x_n = a \implies \lim \lambda_n\cdot ...
18 views

### Example of an uncountable metric space where every point is isolated

I was trying to come up with an example of an uncountable metric space all of whose points are isolated. I've had difficulty thinking of one, has anyone got any nice examples? Just in case: ...
### Show that $\{f_n(x) \}_{n \in \mathbb{N}}$ doesnt converge in M.
Let $n \in \mathbb{N}, f_n(x)=e^{-(x-n)^2}$ and $g(x)=\Bigg\{\begin{array}{ll} 1 & x=0 \\ \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ \end{array}$ You can assume that g is continuous ...