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Jun
6
comment Unit speed geodesics
@Thomas: You're probably right. If $d(x,y)=\inf_{\gamma\in\mathscr{D}(x,y)}L_{d}(\gamma)$ for all $x,y\in X$: where $L_{d}$ is the length-function induced by $d$ and $\mathscr{D}(x,y)$ is the collection of those $\gamma\in X^{[0,1]}$ that are continuous, $\gamma(0)=x$ and $\gamma(1)=y$ and $L_{d}(\gamma)<\infty$ - we say that $(X,d)$ is a length space. In other words, if the distance of any pair of points is obtained as infimum over the lengths of rectifiable paths that join the points.
Jun
6
accepted Unit speed geodesics
Jun
6
revised Unit speed geodesics
added tag
Jun
6
accepted Compactly supported function?
Jun
6
asked Unit speed geodesics
Jun
5
awarded  Civic Duty
Jun
5
comment Cross product of paths is a path with length?
By saying that $a(t)$ is a path with length in $[c,d]$, do you mean that the length of $a(t)$ is a real between $c$ and $d$?
Jun
5
comment Is Koch snowflake a continuous curve?
If the convergence is uniform, then a limit of continuous functions is continuous.
Jun
5
comment (Un-)Countable union of open sets
@DaveL.Renfro. True, I didn't somehow even think about that :-) So $T_{1}$ would be the least requirement for this construction.
Jun
5
comment About a continuous function
Yeah, that does the trick. +1 for nice answer.
Jun
5
comment About a continuous function
Why is $f(t,x)=\max_{y\in X}f(t,y)$?
Jun
5
comment About a continuous function
Are you sure that $m$ is well defined? Should there be $\sup_{x}$ instad of $\max_{x}$? Or are you assuming that $X$ is compact?
Jun
5
comment (Un-)Countable union of open sets
@DaveL.Renfro: can you give some hints how to prove this?
Jun
5
comment Limit of $\frac{x^{x^x}}{x}$ as $x\to 0^+$
Maybe the last equality should be $...=-2\cdot 0\cdot 1=0$ instead?
Jun
4
comment Continuous function on metric space
@JasonDeVito: You probably mean that the sum equals $0$?
Jun
4
comment Continuous function on metric space
@Hiperion: Disjointess is not enough (but it is required), you have to use the property that $A$ and $B$ are closed. Hint: $\bar{A}=\{x\in X:d(x,A)=0\}$, where $\bar{A}$ is the closure of $A$ in $X$.
Jun
4
revised Continuous function on metric space
edited tags
Jun
4
comment Checking for completeness of $\mathbb{R}$ with metric defined by $d_1(x,y) =\mid e^x - e^y \mid$
@srijan: yes, that's correct.
Jun
4
comment Point set topology question: compact Hausdorff topologies
You probably mean "tau", which is \tau and looks the following: $\tau$.
Jun
4
comment Checking for completeness of $\mathbb{R}$ with metric defined by $d_1(x,y) =\mid e^x - e^y \mid$
For $d_{1}$, note that you must start with an arbitrary Cauchy sequence in $(\mathbb{R},d_{1})$ and not in the standard metric. The current argumentation does not show that $d_{1}$ is complete.