I came across this exercise, the main problem for me are the restrictions, i need to find examples for maps $f: \mathbb {R}^2 \rightarrow \mathbb {R}^2$ or subsets of them such that $f$ is only one or two of the three at the same time. (I.e $f$ is open but no closed nor continuous) But i don't know How to build such examples, as i also don't see the kind of patologies i should be looking for
I'm self studying J. Lee's Introduction to Topological Manifolds, and after doing all other exercises on chapter 2, I can't seem to come with the proper counterexamples for this one.      For each of the following properties, give an example consisting of two subsets $X, Y \subset \mathbb{R}^2 $, both considered as topological spaces with their Euclidean topologies, together with a map $f: X \rightarrow Y$ that has the indicated property.         Open but not closed/continuous   Closed but not open/continuous    Continuous but not open/closed         etc... (all cases)   I think the reason for my difficulty is a lack of geometric intuition of what open, closed and continuous maps look like as deformations of shapes in $\mathbb{R}^2$ (I can construct counterexamples from artificial made up spaces).   I tried squashing balls and 'pumping space' into lines, as well as joining strips in annulus shapes, but all the examples I seem to come up with are either too well behaved (i.e. all 3) or too misbehaved.   Instead of a case by case answered I would prefer a couple worked out cases and some intuition on how to visualise open / closed maps in this context.
Why must a computer have a BIOS? The  software is built into the PC, and is the first code run by a   PC when powered on ('boot firmware'). When the PC starts up, the first   job for the BIOS is the power-on self-test, which initializes and   identifies system devices such as the CPU, RAM, video display card,   keyboard and mouse, hard disk drive, optical disc drive and other   hardware. The BIOS then locates boot loader software held on a   peripheral device (designated as a 'boot device'), such as a hard disk   or a CD/DVD, and loads and executes that software, giving it control   of the PC.[2] This process is known as booting, or booting up, which   is short for bootstrapping.   Which requisites determine that a computer system must have a BIOS installed on it?  Why can't the Operational System actually do all of the BIOS work? Fibonacci Sequence and series limits How can I figure out what's slowly eating my HD space? To prevent overfitting people people add a regularization term (proportional to the squared sum of the parameters of the model) with a regularization parameter $\lambda$ to the cost function of linear regression. Is this parameter $\lambda$ the same as a lagrange multiplier? So is regularization the same as the method of lagrange multiplier? Or how are these methods connected? Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$ Show a connected metric space (X,d) is $\epsilon$-chainable for $\epsilon &gt;0$       $\epsilon$-chainable Definition      (X,d) is $\epsilon$-chainable if given any two points $a,b \in X$, $\exists$ a finite set $\{ x_0,x_1,...,x_n\}$ of points in X with $x_0 = a$, $x_n=b$ and $d(x_{n-1},x_n)&lt;\epsilon$, $\forall$ n      Clopen Definition      A set is clopen $iff$ its boundary is empty      Connected Metric Definition      Let $(X,d)$ be a metric space. It is connected $iff$ it has no proper subsets which are clopen      The following are equivalent:         $X$ is connected   The only subsets of $X$ that are clopen are $\emptyset$ and $X$   $\nexists$ non-empty, disjoint, open sets $A$ and $B$ such that $X=A\bigcup B$   $\nexists$ non-empty, disjoint, closed sets $A$ and $B$ such that $X=A\bigcup B$         Theorem      Let $(X,d)$ be a metric space, and let $S$ be any subset of $X$. Then $S$ is closed $iff$ $S^c$ is open.     So how would someone show this exactly?  Let the set $A$ be the set of all elements in $(X,d)$ that can be reached from $a\in X$ in a finite number of steps with length less than $\epsilon$.  Does this mean that there should be no isolated points? (an isolated point would be at distance infinity from any other point, or distance zero from itself, since $\epsilon &gt;0$ we don't want this?)  Knowing that $X$ is connected $\implies$the only subsets of X that are clopen are $\emptyset$ and $X$  Also, by construction of A;   A is the set of all $\epsilon$-chains between any element and $a$  And, so $A^c = \emptyset$ if $A$ was $\epsilon$-chainable (because A has to have a path from all elements to a, for this to be true, the complement wouldn't have anything in it)  Leaving only $A$ as the remaining clopen set  Does this make any sense? Or should an alternative be used, and how to formalize it.  Thanks. When are "if" and "whether" equivalent? Are if and whether equivalent in sentences like the ones below?     How to determine if my saddle is too high?   How to determine whether my saddle is too high?         We should check if everything is okay now.   We should check whether everything is okay now. modulo version of the quadratic formula and Euler's criterion cp - cannot create regular file : permission denied I want copy to /lib/udev but gives error  cp -f /tmp/ok_pcscd_hotplug.sh /lib/udev  cp:cannot create regular file /lib/udev/ok_pcscd_hotplug : Permission denied   I am root , but I can't understand . what can I do ? 'An avalanche of mail'? - it should be 'mails', shouldn't it? I came across this example on WordWeb while going through the meaning of the word ''.     the program brought an avalanche of mail (Just like "a group of man men").   I'm quite sure it should be mails. Kindly confirm. At the beginning of  we see that Logan (a.k.a. Wolverine) and Colonel William Stryker were working together by the end of the Vietnam War.   For 5 years after the war, he lives in Canada with a girlfriend. But in  we see that Logan was part of some gang at the end of the Vietnam War.   My question is which of these are X-Men movie canon? Does separability imply the Lindelöf property? Does separability imply a sort of Lindelöf property?  Since I can't prove this fact I'm beginning to think that my conjecture is false.  Intuitively, $\mathbb{R}$ has a countable subset $\mathbb{Q}$ which is used to form a countable basis for $\mathbb{R}$ with the usual topology and prove the Lindeöf property. I'm trying to figure out the name of a movie I saw on TV as a kid, in the mid-to-late 90s, in the US, in English, on daytime TV.  I remember it being kind of Starship Troopers-esque, but everyone had these glowing lights in their chests.  They glowed in different colored patterns, and I think it had something to do with whether you were attracted to someone or not (I was probably like 7 when I saw it, I didn't really understand the context of that part yet). If $\int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then every line $y=mx$ Intersect $ y^2+\int_{0}^{x}f(t)dt=2$ If $f$ is a continuous function with $\displaystyle \int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then show that every line $y=mx$ Intersect the curve $\displaystyle y^2+\int_{0}^{x}f(t)dt=2$  $\bf{My\; Try::}$ Put $y=mx$ in $$\displaystyle y^2+\int_{0}^{x}f(t)dt=2\;,$$ We get $$\displaystyle m^2x^2+y^2+\int_{0}^{x}f(t)dt=2.$$ We can write it as $$\displaystyle 2\int_{0}^{x}2m^2tdt+\int_{0}^{x}f(t)dt=2$$ So $$\displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2 =0$$ Now Let $$g(x) = \displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2.\;,$$ When $x\rightarrow \infty\;,$ Then $g(x)\rightarrow \infty.$ (Where $g(x)$ is an Continuous function.) Now I did not Understand How can i solve that, Help me Thanks These badges were only recently added to StackExchange. I would like to know if there is any way I can find out how many more questions I should ask to get these badges.   For e.g. like the counters for the review-related badges: Programmatically obtain the phone number of the Android phone I made a character, and its one of my first times texturing, and I saved my character into a new folder, and when I came back to the model ALL the textures were pink, I used to know what to do, but I stopped texturing, how do I bring my textures back? Returning from a finally block in Java I was surprised recently to find that it's possible to have a return statement in a finally block in Java.  It seems like lots of people think it's a bad thing to do as described in ''. Scratching a little deeper, I also found '' which shows some pretty horrible examples of other types of flow control in finally blocks.  So, my question is, can anyone give me an example where a return statement (or other flow control) in a finally block produces better / more readable code? How can I do a Cache Refresh in Google Chrome? I don't know what exactly it's called, by cache refresh I mean, refresh the page after clearing its cache. I don't want to clear the entire browser cache.  I can't seem to cache refresh my pages. In Firefox, I know it to be Shift+Refresh.  In Chrome, I've tried Ctrl+R, Ctrl+Refresh, Alt+Refresh, Shift+Refresh but none of them work. Scatterplot with too many points Insert a paragraph into a list that uses the whole text width