My intuition tells me that the only connected 1-dimensional topological manifolds are the real line $\mathbb{R}$ and the circle $S^1$.   Is this true?   If yes, is it possible to prove it from first principles, or is it something that needs some highly technical theorems?  If no, do we have an example of connected 1-dimensional manifold not homeomorphic to $\mathbb{R}$ or $S^1$?
I recall having heard somewhere that the only 1-manifolds (second countable, Hausdorff, connected spaces locally homeomorphic to $\mathbb R$) are $\mathbb R$ and $S^1$. Is this true? If so, is there a reasonably elementary proof?
How can a company use money from stock investors when they are constantly being bought and sold? I am interested in the mechanisms which a company can use the money that investors give them when they buy a public stock since they are being bought and sold constantly. How can a company spend other people's assets when the shareholders still own their stocks and they still retain a liquid value? To me it seems like stocks are a case of "having your cake and eating it too". How to determine $\int x^2 e^{-x^2}\mathrm dx$? How do I change the plymouth bootscreen? With regards to Plymouth --  Are there alternative boot screens available?  What's the easiest way to change the boot screen? Double booked - User bookmarking question recorded twice In this  one user is shown as having bookmarked  twice:  This doesn't appear to be the same question as:  or  though it's certainly possible that the cause is the same. Tikzexternalize not compatible with Miktex 2.9 abntex2 package Does umount calls sync to complete any pending writes We have BBB based custom board with kernel 3.12 running on it.  I have doubt regarding umount and &amp; sync.  Lets say a script is umounting a partition, Does it require to run sync command before umount to complete pending writes. ? I asked a  recently, looking for examples of an online tool. Per the , I answered my own question with a few examples that I had already found. I did this so that these examples would be included in the answers, along with the rest.  The question has received only down votes and insults saying that this shouldn't be done.   Should I not answer my own questions? Why do diamonds shine? Running two MySQL servers that use the same data directory How does editing work? It is possible for users to edit an existing post.   How do edits work? Who may edit a question or answer? How can you tell what has been changed between edits?      For more information, see "" in the .     Related: Assuming I'm paying long term capital gains tax, what is the break even point where it is more worthwhile to switch from one fund to another?  For (a hypothetical) example, if I have a fund with an expected return of 3.5%, what would be the minimum expected return of another fund to make it worthwhile to pay the capital gains tax and switch to the other fund?  (I'm assuming the answer to this question is an equation involving investment horizon, capital gains tax rate, and my current taxable gains) How should I successfully hibernate Ubuntu 16.04?  I've tried almost every solution on the web but non of them was actually helpful. I found my problem almost completely similar to I am making a fudge recipe that calls for 1cup whole milk and 5 cups sugar. It also uses 2 sticks of butter and 25 marshmallows. I'm wondering if I can omit the milk and cut back on the sugar by using sweetened condensed milk. Check if table exists in SQL Server I would like this to be the ultimate discussion on how to check if a table exists in SQL Server 2000/2005 using SQL Statements.  When you Google for the answer, you get so many different answers. Is there an official/backward and forward compatible way of doing it?  Here are two possible ways of doing it. Which one among the two is the standard/best way of doing it?  First way:  IF EXISTS (SELECT 1             FROM INFORMATION_SCHEMA.TABLES             WHERE TABLE_TYPE='BASE TABLE'             AND TABLE_NAME='mytablename')     SELECT 1 AS res ELSE SELECT 0 AS res;   Second way:  IF OBJECT_ID (N'mytablename', N'U') IS NOT NULL     SELECT 1 AS res ELSE SELECT 0 AS res;   MySQL provides the simple   SHOW TABLES LIKE '%tablename%';    statement. I am looking for something similar. When is the graph of the divisor relation on $[n]$ planar? A recently-deleted question asked to show that if $G(n)$ is the graph on $[n]=\{1,\dots,n\}$ where $x$ and $y$ are adjacent if $x|y$ or $y|x$, then $G(n)$ is planar.  This is false if $n\geq150$, for the subgraph induced by $\{2,3,5,90,120,150\}$ is isomorphic to $K_{3,3}$.  My question is, what is the largest value of $n$ such that $G(n)$ is planar?  Of course, various software packages would make short work of this, but I'd be interested in a verbal argument, if someone can give one. How can I disable the Windows key from opening the Activities overview? How can I disable the Windows key (or Super) from showing the GNOME Shell Activities overlay? Potential energy of an infinitesimal length of elastic rod Is there any software or hardware which lets you stop, slow down, speed up or even reverse time? Obviously I'm talking about time in terms of the PC clock rather than real time.  We were testing an application we've developed at work by setting the clock forward and back to simulate different scenarios and I started thinking how useful it would be if you could adjust the rate(?) of the system clock with finer control. So you could make a minute pass in a second or a day pass in 30 seconds and watch how the program you're developing copes with changes in date and time.  Is there any such software available?   I'd be interested to hear if anyone knows of any software or hardware which can let you do some or all of the above. How do I mount shared folders in Ubuntu using VMware tools? Experts. I've successfully installed VMware tools for Ubuntu. Everything seems to work fine, but shared folders were not mounted automatically. How do I get them to work?  If I run vmware-hgfsclient in terminal, I get the list of shared folders, but ls -l /mnt/hgfs is empty. Actually there's no hgfs dir in /mnt. I know I should probably use the vmware-hgfsclient tool, but I realy don't know how.   P.S. I wouldn't ask if I could understand the vmware-hgfsclient help I've read.   Any suggestions? The screenshot below says it all, really. The object is offset in both the X and the Z axes, though only the X axis says 1. No other modifiers are on it, no constraints, no locked axes, and I can't think of anything else that would affect this. How can I make it stop doing this?