Good night, someone could help me with some interesting applications of this important theorem and also I could say how I draw a dense subset in the complete space used in the theorem. Thanks for your help
I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications that confirm its neatness and/or power.  Here's the theorem (with proof) and two applications:    (Baire) A non-empty complete metric space $X$ is not a countable union of nowhere dense sets.  Proof: Let $X = \bigcup U_i$ where $\mathring{\overline{U_i}} = \varnothing$. We construct a Cauchy sequence as follows: Let $x_1$ be any point in $(\overline{U_1})^c$. We can find such a point because $(\overline{U_1})^c \subset X$ and $X$ contains at least one non-empty open set (if nothing else, itself) but $\mathring{\overline{U_1}} = \varnothing$ which is the same as saying that $\overline{U_1}$ does not contain any open sets hence the open set contained in $X$ is contained in $\overline{U_1}^c$. Hence we can pick $x_1$ and $\varepsilon_1 &gt; 0$ such that $B(x_1, \varepsilon_1) \subset (\overline{U_1})^c \subset U_1^c$.   Next we make a similar observation about $U_2$ so that we can find $x_2$ and $\varepsilon_2 &gt; 0$ such that $B(x_2, \varepsilon_2) \subset \overline{U_2}^c \cap B(x_1, \frac{\varepsilon_1}{2})$. We repeat this process to get a sequence of balls such that $B_{k+1} \subset B_k$ and a sequence $(x_k)$ that is Cauchy. By completeness of $X$, $\lim x_k =: x$ is in $X$. But $x$ is in $B_k$ for every $k$ hence not in any of the $U_i$ and hence not in $\bigcup U_i = X$. Contradiction. $\Box$    Here is one application (taken from ):  Claim: $[0,1]$ contains uncountably many elements.  Proof: Assume that it contains countably many. Then $[0,1] = \bigcup_{x \in (0,1)} \{x\}$ and since $\{x\}$ are nowhere dense sets, $X$ is a countable union of nowhere dense sets. But $[0,1]$ is complete, so we have a contradiction. Hence $X$ has to be uncountable.     And here is another one (taken from ):  Claim: The linear space of all polynomials in one variable is not a Banach space in any norm.  Proof: "The subspace of polynomials of degree $\leq n$ is closed in any norm because it is finite-dimensional. Hence the space of all polynomials can be written as countable union of closed nowhere dense sets. If there were a complete norm this would contradict the Baire Category Theorem."
Where can I find sugar cane? I played several hours a started three different worlds in Minecraft, but I have never found any sugar canes. I read on Minecraft wiki to search near water but still no luck. Do you have more hints? Is it possible to re-create an environment that allow sugar canes to naturally grow? 555 Monostable Timer delay I have built numerous 555 circuits over the years. I recently had to built a Monostable 555 cct. as per this site :    The cct. is used in a motor vehicle. When the remote control is pushed to open the doors, it sends positive 12V to this Monostable cct. to initiate it and a 3 Amp Power Transistor pulses the siren momentarily to indicatse "unlocking" by audible means.   The issue I have is that Rt and Ct used to time the output pulse is not short enough. I replaced these components to the smallest values I could get, but the output is still "too long". Teh siren makes a "chooow" sound instead of "chow". Sounmds too unprofessional or after-market.  Any ideas to get a short pulse (nearly a few micro seconds) out of this old 555 cct. ? Is it possible to encrypt a message in such a way that multiple keys can independently decrypt it?  In other words:  Having knowledge of $key_1, key_2 ...key_N$, which are different symmetrical keys, is it possible to generate a ciphertext that can be independently decrypted using either $key_1$ or $key_2$ or $key_N$, in such a way that the length of the ciphertext is less than directly proportional to $N$? Is the universe finite or infinite? I thought the universe was finite, but then I read this:    And they seem to assume it is infinite. So which is it? Stopping resetting of linked locators in manipulate Does iOS support a built-in firewall like macOS does? I don't see anything about it in Settings. Print lines between (and including) two patterns I want to start grepping from the lines that has CK at the end of line and stop grepping when the line has D at the end. 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