Consider a countable set $X$ with cofinite topology.  the claim is that $X$ is second countable.  The simple argument is as follows :  $\color{red}{\text{Since $X$ is countable, the number of finite sets in $X$ are countable.}}$ Hence the number of open sets in $X$ are also countable. Since the topology itself is countable, the space is second countable.      Question: Why should it be true that the number of finite subsets of a countable set is countable?
Show that the set of all finite subsets of $\mathbb{N}$ is countable.   I'm not sure how to do this problem. I keep trying to think of an explicit formula for 1-1 correspondence like adding all the elements in each subset and sending that sum to itself in the natural numbers, but that wouldn't be 1-1 because, for example, the set {1,2,3} would send to 6 and so would the set {2,4}.  Multiplying all the elements in each subset and sending that product to itself in the natural numbers wouldn't work either since, for example, {2,3} would send to 5 and so would the set {1,5}. Any ideas?
Do the automatic success and failure rules apply to skill checks? We had a rule check tonight that we couldn't answer. On the Pathfinder wiki I could only find info on natural 20s and 1s in regards to attack rolls and saves in combat.  Do they also apply to skill checks? Is there an Attribute I can use in my class to tell DataGridView not to create a column for it when bound to a List I am trying to solve a problem from Stanley's book, it says:  Fix $k,n \in \mathbb{P}$. Show that: \begin{align} \sum a_1 a_2 \cdots a_k = \binom{n+k-1}{2k-1} \end{align} where the sum ranges over all compositions $(a_1 , a_2 , \ldots , a_k)$ of $n$ into $k$ parts.  I am trying to reason like this:  we need to find the coefficient $c_n = \sum a_1 a_2 \cdots a_k$ from this generating function --  \begin{align} \sum_n c_n x^n &amp;= \sum_n \sum a_1 a_2 \cdots a_k x^n \\  &amp;= \sum_n \sum a_1 a_2 \cdots a_k x^{a_1 + a_2 + \cdots + a_k}\\ &amp;= \sum_n \sum a_1x^{a_1} a_2x^{a_2} \cdots a_kx^{a_k} \end{align}  after that, I have no clue, how do I solve this ?   moreover, what is the range in the inner sum ?    If we consider Mark Riedel's answer, and assume $n=4$, $k=2$; then the sum will be  \begin{align} \sum (z + 2z^2)^2 = z^2 + 4z^3 + 4z^4 \end{align}  On the other hand the compositions will be $(1,3), (2,2), (3,1)$, therefore the above sum will be counted as:  \begin{align} (1.3)z^{1+3} + (2.2)z^{2+2} + (3.1)z^{3+1} &amp;= 1z^1.3z^3 + 2z^2.2z^2 + 3z^3.1z^1\\  &amp;= 3z^4 + 4z^4 + 3z^4 = 10z^4 \end{align}   what's going on? what am I missing? I designed a poster in CorelDraw X5 for printing purposes. I was told by the Printshop guy to use colors only in CMYK. For the most part I did, I don't know why I missed it but I used one color in RGB. I liked this color so much that I've used it at many places. So, current situation is that some colors are in RGB and some are in CMYK. So, I want to convert all RGB to CMYK and thought the problem would be solved. But now I'm actually facing a lot of trouble. The colors are not at all same. I hate the colors that it is showing me.  This is the color I used in RGB:    When I changed the "Model" to CMYK, the color it shows is really different. As you can see in the picture below, it's too dull:    The RGB values of both colors are not the same. What can I do to get the same color in CMYK? I did a little bit of reading here and there. They say, that you can't get same. In that case, what is the closest color that I can get? Mark read notifications as unread In the Android app, can we have some option to mark a read notification to unread? I know that this option is not available in the web version of Stack Exchange sites. But sometimes I open the notification in the app and it opens the post, but sometimes I can't see the exact reply to my comment or where my name is mentioned using the app.   And I want those comments to be read from the web version of the Stack Exchange site. Sometimes I forget about replying to a comment since the notification is already marked as read. So I want to mark the notification as unread and have it as a new notification displayed in the mobile app and the web version too. When a land becomes creature, can you kill it? How do I fix inconsistent Textarea bottom margin in Firefox and Chrome? Why won't attribute modifiers add up on my armor? How do I get rid of eggs? Why is the light reflected at the same angle from mirror? From the school physics I know that the material objects bounce from the plane surface at the same angle, losing some kinetic energy. In the same school I was taught that the light (and waves in general) obeys this principle too.  Obviously, in the case of light the plane surface should be a perfect mirror. But I can't understand how should this work from the quantum point of view. Let's assume that our mirror consists of a single silver atom.  Why should the electrons of this atom re-emit consumed photons at some specific angle? Definite integral, quotient of logarithm and polynomial: $I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$ Can mobs go through doors? I was wondering if mobs can go through doors or not. It's the difference between me just putting up a door to my house, or putting up dirt instead. Can you explain the sentence structure 'In a hole in the ground there lived a Hobbit'? Why put the verb before the subject? The opening sentence to The Hobbit by J.R.R. Tolkien reads,     In a hole in the ground there lived [verb] a hobbit [subject].   I wonder if there are accepted stylistic purposes for such a structure.  When is it natural, and when is it unnatural? Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$ I am trying to calculate the final velocities of two equal mass 2-dimensional circles after an elastic collision. I have tried to figure it out using formulas I know from high school physics, but nothing seems to work. My known variables are:   initial x and y velocities for each circle x and y distances between the centres of the circles at the time of the collision   The variables I would like to solve for are:   final x and y velocities for each circle   Here are diagrams showing all the variables: Does order matter when casting a cantrip with a casting time of a bonus action and another spell? When casting a spell as a bonus action, you can't cast another spell on the same turn except a 1-action cantrip - can the order be "reversed"? Can you cast a cantrip first as a bonus action, then any other spell? If not, would it be broken to allow it?  I'm thinking a Sorcerer casting a Quickened true strike, then casting a leveled spell requiring a touch attack. Seems like a great use of the cantrip + Metamagic feature, but it seems like it is not allowed. Which is correct: "he don't" or "he doesn't"? How to roll perception checks for characters who aren't actively looking without arousing suspicion? This applies to all d20 systems, but with an example from the from the Pathfinder  (emphasis mine):  Stonecunning: Dwarves receive a +2 bonus on Perception checks to potentially notice unusual stonework, such as traps and hidden doors located in stone walls or floors. They receive a check to notice such features whenever they pass within 10 feet of them, whether or not they are actively looking.  In this case, does the GM roll the check in secret or ask the player roll the check? If the GM asks the player to roll for the check, what stops situations like:  GM: (interrupting) &quot;Gimli, Roll a perception check.&quot; Dwarf player: &quot;Oh, it's a natural one, I guess I don't see anything.&quot; Human player: &quot;Okay, I'll roll a perception check for anything unusual in this room. Nice! 19. How about 19, does 19 work for you?&quot; GM: &quot;Gimli, everything looks kosher to you. Aragon, you notice a rock coloured tarp on the ground here, covering what looks like a pit trap.&quot;  In this case, by telling the dwarf to roll his &quot;free&quot; perception check, he's telegraphed to the players that there is something to notice. How can this be fixed, without bluffing checks for nothing in particular to throw off the signal to noise ratio of similar checks? Seems that I just proved $2=4$. Why the following example prints "0" and what must change for it to print "1" as I expected ?  #include &lt;iostream&gt; struct base {    virtual const int value() const {       return 0;    }    base() {       std::cout &lt;&lt; value() &lt;&lt; std::endl;    }    virtual ~base() {} };  struct derived : public base {    virtual const int value() const {       return 1;    } };  int main(void) {    derived example; }