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1d
answered Orders with no anti-symmetry requirement?
2d
comment Image of a continuous function
@MikeMiller What about the function $(1-x)\sin(1/x)$? That maps $(0,1]$ to $(-1, 1)$, and can clearly be modified to map into $(0,1)$.
2d
answered Proof of Pythagorean theorem without using geometry for a high school student?
2d
revised $G =(V,E)$ is $k$-connected ($k \geq 2$), prove that for every subset $S \subseteq V $, |S|=k there exists a cycle in $G$ that goes through all of $S$
Changed 'group' to 'subset' (the term 'group' has a very specific meaning in mathematics, and I don't think that's what's meant here)
Aug
1
answered Show that $\mathbb{R}^2/{\sim}$ is homeomorphic to the sphere $S^2$.
Jul
30
awarded  Nice Question
Jul
27
revised What is a homology of chain complex?
added 445 characters in body
Jul
27
comment What is a homology of chain complex?
@AllenCho Nice question! The answer is YES, and the reason is singular homology. Singular homology is quite similar to simplicial homology, but it can be calculated for every topological space. That eliminates any difficulties arising from non-triangulable manifolds. If simplicial homology were our only tool, you are right that this would be a problem.
Jul
27
answered What is a homology of chain complex?
Jul
26
answered Connected components and showing subsets are equal
Jul
26
comment Show $X$ is simply-connected given properties of two subsets
What if $X$ is the disjoint union of two simply connected manifolds? Then it is not simply connected (since path connectedness is part of simple connectedness), but we can write $X=U\cup V$, where $U,V$ are the two connected components, which are connected and simply connected: then $U\cap V=\emptyset$, which is connected.
Jul
26
answered Proof attempt for collection of all open intervals being a basis of $\Bbb R$ with the standard topology
Jul
25
comment Compose $(1243)$ and $(5)$
What do you mean by $(5)$? The convention with disjoint cycle notation is that we ignore $1$-cycles - so $(5)=(1)(2)(3)(4)(5)=\text{id}$.
Jul
23
comment Differentiate the Function: $y=x^x$
On the last part, close, but not quite - $(x^a)'=ax^{a-1}$, and besides, $x^x\ln(x)+x.x^x=x^x(\ln(x)+x)$, not $x^x(\ln(x)+1)$.
Jul
21
comment Cubic function: Cardano's method
@TommyLassa Good question, and the surprising answer is that you can't in general. For some examples, such as what you get from expanding out $(x-1)(x-2)(x-3)$, say, you have nice rational solutions, but in certain cases (look up casus irreducibilis), it is impossible to get a closed-form radical expression for the solutions of a cubic without moving outside the real numbers even when the coefficients of the cubic and all three of its roots are real.
Jul
21
answered Cubic function: Cardano's method
Jul
21
answered Show that $A\subseteq B\implies A^{\circ} \subseteq B^{\circ}$ in a different way.
Jul
20
comment How do I show that we can't write $N=114^n-1$ as sum of $3$ squares for all natural number $n>2$?
What computations did you carry out on Wolfram|Alpha? Can you reproduce them for us?
Jul
15
answered Why do we say “radius” of convergence?
Jul
14
answered In $(X,d)$ metric space, an intersection of finite families of open sets is open.