# Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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### Why is this a dense set? (Lemma 6.13, Lee ISM)

In Lemma 6.13 of Lee's 'Introduction to Smooth Manifolds' Lee is showing that the images of two functions $\kappa, \tau$ are subsets of measure zero in $\mathbb{RP}^{N-1}$. It follows then that their ...
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### prove every open interval in R is open set [closed]

Prove that $(a,b)$ is open set in $\mathbb R.$
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### Is there something missing in this exercise?

Let $n \in \mathbb{N^{*}}$ and $||\cdot||$ be a $\mathbb{R^n}$ norm. Let $F \subset \mathbb{R^n}$ be a closed set. I have to prove that $F_{\epsilon}:={\{x\in \mathbb{R^n}:d(x,F)\leq \epsilon\}}$ is ...
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### Embedding/map that respects boundaries?

What do you call an embedding (or map in general) from a manifold to a manifold $i: M \rightarrow N$, such that $i(\partial M) \subseteq \partial N$? Is there a difference, if we define it as a ...
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### Base in topological space

Let $B_1$ and $B_2$ be the basis in the topological space $(X,\tau)$. Will $B_1\cup B_2$ and $B_1\cap B_2$ be the base? I think intuitively it would be $B_1\cup B_2$. But $B_1\cap B_2$ may not be ...
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### About Density and continuous and open function.

I've seen a proposition like that Proposition: If $f:X\longrightarrow Y$ is an open and continuous function and $D$ is dense in $Y$, then $f^{-1}(D)$ is dense in $X$. Is that proposition correct? if ...
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### Prove that a set $Y$ in a metric space $(X, d)$ is open if and only if it contains none of its boundary points.

I will preface this question by saying that I realise that there are a lot of similar questions to this one, which have already been answered on this site, but none, which are exactly this question. ...
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### If $L$ is a straight line then what is the topology that $L$ inherits as a subspace from $\mathbb{R}_l \times \mathbb{R}$

If $L$ is a straight line then what is the topology that $L$ inherits as a subspace from $\mathbb{R}_l \times \mathbb{R}$ and as a subspace of $\mathbb{R}_l \times \mathbb{R}_l$ . I know that the ...
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### Is the following an open cover for the set $K = \{0\}\cup \{1/n\mid n \in \mathbb{N}\}$?

I know it has been asked to death on this site how to prove that $K = \{0\}\cup \{1/n\mid n \in \mathbb{N}\}$ is compact. I would like to not be spoiled about the proof as IMO my question is ...
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### Second Countability Implies Separability

Munkres states explicitly that: Theorem 30.3: Suppose that $X$ has a countable basis. Then: (b) There exists a countable subset of $X$ that is dense in $X$. The following proof he gives is ...
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### Proving semiregularity of a topological space.

I'm working in some exercise of the book Extensions and Absolutes of Hausdorff Spaces by Porter and Woods and I'm stucked in the proof. The exercise is $7$H(2): Let $X$ be a regular space, $Y$ a set ...
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### Canonical LF topology

Wikipedia: The space of test functions usually consists of smooth functions with compact support that are defined on some given non-empty open subset $U\subseteq \mathbb {R} ^{n}$. ...
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### X is a connected and compact Hausdorff space with some properties.Prove that X is homeomorphic to $\mathbb{S}^1$.

$X$ is a connected and compact Hausdorff space,and for all $x \in X$,there is a open neighborhood of $x$ homeomorphic to $(-1,1)$.Prove that $X$ is homeomorphic to $\mathbb{S}^1$. At first,I replace X ...
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### Show that if $A$ is a basis for a topology on $X$,then what will be the topology generated by $A$

Show that if $A$ is a basis for a topology on $X$,then the topology generated by $A$ equals the intersection of all the topologies on $X$ that contain $A$.Prove the same if $A$ is a sub basis. My ...
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### the minimal uncountable well-ordered set $S_\Omega$ and the sequence lemma (example 3, sec 28 in Munkres topology)

First How do prove that $S_\Omega$ satisfies the sequence lemma? Munkres says 'you can readily check', but it is not easy for me. Second How does the fact that there is no sequence of $S_\Omega$ ...
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### Baby Rudin Definition 2.18 (i) : Bounded set

I am trying to self study Rudin's Principles of Mathematical Analysis and I have been stuck on the definition of bounded sets stated in the book: $E$ is bounded if there is a real number $M$ and a ...
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### Partial Converse of Uniform Limit Theorem from Munkres' Topology

Here is the question given in the book Topology written by James R. Munkres. Prove the following partial converse of the uniform limit theorem: Let $f_n : X \to \mathbb{R}$ be a sequence of ...
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### Let $A \subseteq X$ be closed and $X$ compact with $A \neq \emptyset \neq X$. Is $C \cap \partial A \neq \emptyset$ for each component $C$ of $A$?

I know the result is true if $X$ is additionally assumed to be connected. Is the claim false without the connectedness? I can't think of a counter-example.
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### Conceptual proof of Riemann–Hurwitz formula

Lately I came across the very interesting Riemann–Hurwitz formula. I believe I understand the claim of the formula, but I do not understand well why this formula is true. I am looking for some ...
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### How does rotating a circle create a spherical surface without leaving “gaps”?

EDIT: Agreed, this isn't a well formed question. But responses below have at least given me a different way to think about it. First post here because I'm not a math guy, but I have a feeling the ...
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### extension to a continuous function

I self-studying Aluffi chapter 0. I need the following result to prove a exercise. Assume that K is compact topological space. Fix a $p \in K$. Assume that $g : K \rightarrow \mathbb{R}$ is continuous ...
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### On the basis for quotient topology

There is some discussion on this topic already in Basis for the Quotient Topology. But I had some other questions. If I start with a basis $\mathcal{B}$ in the original topological space X and ...
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### Homeomorphism between two disks with a hole

I need to show that $\mathbb{D}^2\setminus B(x_1,r_1)$ and $\mathbb{D}^2\setminus B(x_2,r_2)$ are homeomorphic. I have tried with a Möbius transformation of the disk but I can't show that it sends the ...
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