# Tagged Questions

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### Compact Set: Cover by Merely Neighborhoods

Disclaimer: This thread is just a record of thoughts and written in Q&A style. A subset is compact if every open cover admits a finite subcover. What if one replaces open covers with covers by ...
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Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
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### What is the definition of Lindelöf space?

My definition for "countable set" is a set with the cardinal $\aleph_0$ and "at most countable set" is a set $A$ such that $|A|≦\aleph_0$. Till now, my definition for Lindelöf space is a topological ...
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### Definition verification from two different books?

In Kaplansky's Set Theory And Metric Spaces, he mentions a useful example of a neighborhood of $x$ is a closed ball with center $x$. However, one of the theorems in baby Rudin is "Every neighborhood ...
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### Topology generation

What does it mean for a topology to be generated? For example $X=\mathbb{R}$ be topology generated by $[a,b)$. Isn't the topology a collection of open sets? $[a,b)$ is not open though.
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### A function “extends” to the cone on X

I have the following statement: A map $f : X \rightarrow Y$ is nullhomotopic if and only if it extends to the cone on $X.$ My problem is that I have no idea what "extends" means in this statement (I ...
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### If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous?

If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous? I'm looking at a proof where they only show that $f$ is continuous and 1-1. Then I looked ...
the definition of Dedekind cut by initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B \neq \emptyset$, $B$ is Dedekind cut of $A$ under $\preceq$ ...