I'm reading Munkres topology and in section 17 when he introduces limit points, he goes from talking about open sets and elements to points. I can't find where he actually defines "point". Is a point an open set? Or is a point a singleton? Would the indiscrete topology then have no points?
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$\begingroup$ Points are elements. Limit points are also elements. $\endgroup$Thomas Andrews– Thomas Andrews2025-09-18 00:50:22 +00:00Commented Sep 18 at 0:50
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1$\begingroup$ As a general rule, we talk about elements of a set, and points are elements of a topological space. $\endgroup$Thomas Andrews– Thomas Andrews2025-09-18 00:52:12 +00:00Commented Sep 18 at 0:52
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2 Answers
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If $(X,\mathcal{T})$ is a topological space, then:
- Elements of $X$ are called points
- Elements of $\mathcal{T}$are called open sets
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$\begingroup$ I think this is correct. Please let me know if I'm wrong! $\endgroup$Caleb Jares– Caleb Jares2025-09-18 00:51:38 +00:00Commented Sep 18 at 0:51
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2$\begingroup$ Yes. In general, for a topological space, we don't talk about the elements of the space. Most people would read "elements of a topological space" as points, but we almost always call the elements of $X$ points. $\endgroup$Thomas Andrews– Thomas Andrews2025-09-18 00:53:46 +00:00Commented Sep 18 at 0:53
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2$\begingroup$ And we'd never call an open set an element of the space. We can call it an element of the topology of he space. $\endgroup$Thomas Andrews– Thomas Andrews2025-09-18 00:56:57 +00:00Commented Sep 18 at 0:56
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Let $(X,\mathcal{T})$ be a topological space.
If points and open sets are same then $X=\mathcal T.$ Since, $\mathcal T\subset 2^X$, the power set of $X$, we get $$X\subset 2^X$$ which is true only for some pathetical spaces.
Forexample, if $X=\emptyset$, then $|X|=0$ but $|\mathcal T|=1$, so $X\neq\mathcal T$ in that case too.
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2$\begingroup$ This is incorrect. Consider the one point space $\{\emptyset\}$ with its unique topology. $\endgroup$Steven Clontz– Steven Clontz2025-09-19 00:14:43 +00:00Commented Sep 19 at 0:14
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$\begingroup$ @StevenClontz İ didn't say it is can't be true. I said only absurd. And I did not understand the downvote. I think some haters. $\endgroup$Bob Dobbs– Bob Dobbs2025-09-19 05:36:23 +00:00Commented Sep 19 at 5:36
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1$\begingroup$ More simply: If $X=\mathcal T$ then $X\in\mathcal T=X$ so $X\in X$ contrary to the axiom of foundation. If we drop that axiom, we could have a topological space $(X,\mathcal T)$ where $\mathcal T=X=\{\varnothing,X\}$. $\endgroup$user14111– user141112025-09-19 06:05:58 +00:00Commented Sep 19 at 6:05
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1$\begingroup$ @user14111 Thanks for the comment. Your solution is simpler. The axiom of foundation did not come into my mind. $\endgroup$Bob Dobbs– Bob Dobbs2025-09-19 07:05:51 +00:00Commented Sep 19 at 7:05