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Every hereditarily Lindelöf scattered space is countable

Yes, this is a well-known result. I guess it is contained in one of the books of Juhasz about cardinal functions. In fact, it holds for arbitrary cardinals, namely: Let $X$ be an infinite, scattered ...
• 4,437

• 79.4k
Accepted

Lifting a particular homotopy commutative diagram

Let $h\colon X\times I\to Y'$ be a homotopy that makes the diagram commute, i.e. $h_0$ is equal to going through the upper right corner and $h_1$ is equal to going through the lower left corner of the ...
• 4,029

Defining Real Numbers using Cauchy Sequences

"Naturally, the codomain of this metric space's distance function cannot be the real number system, as that's what we're trying to define and can't presuppose its existence." It is true as ...
• 44.3k

Is $E \cup \{\infty\}$ necessarily homeomorphic to some unit sphere?

The answer Yes, $\tilde{E}$ is always homeomorphic to the unit ball of some $F$. Let $F = \mathbb{R} \oplus_2 E$. Let $S$ be the unit sphere of $F$. Consider the maps  f: S \to \tilde{E}, \begin{...
• 1,832

Punctured plane is homeomorphic to the plane without the unit disk.

Well, you have to use the definition of $f$ to at least witness $f$ is bijective. Then in your proof, you make the following key oversight: you must check $z>0$ ! But you did not. Really, this is ...
• 43.4k
1 vote
Accepted

If $\{K_n\}_n$ is a family of compacts such that $\bigcap_nK_n=\varnothing$, then $\exists N\in\mathbb{N}$ such that $\bigcap_{i\le N}K_i=\varnothing$

This is not true in general. For example, consider $\mathbb{N}$ with the cofinite topology (whose open sets are $\varnothing$ and $U\subseteq \mathbb{N}$ such that $\mathbb{N}\setminus U$ is finite). ...
• 79.4k
1 vote

• 4,029
1 vote

Is there an introduction to uniform structures besides Bourbaki?

Chapter 9 “Uniform Spaces” in General Topology by Stephen Willard.
• 2,493
1 vote
Accepted

Product of closed sets is closed in product topology

You're right that your version would be what results from the top formula, but your professor's version is also a valid formula, even if it doesn't result from the top one (and is the one we want to ...
• 11.2k

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