# 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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### What does “finitely many” mean here? “$B$ is a collection of sets that are the intersection of finitely many sets in [a set of sets]”

Take a set of sets $C$. What does it mean when we say "$B$ is a collection of sets that are the intersection of finitely many sets in $C$?" Does "finitely many" mean "from 1 to n"? With $1$, I ...
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### If a topology over an infinite set contains all finite subsets then is it necessarily the discrete topology?

If we define a topology via open sets then yes, infinite unions of the finite subsets will necessitate the infinite subsets also part of the topology giving us the discrete space; however, if we ...
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### Nets, dense subsets and continuous maps

Let $X$ and $Y$ be topological spaces, with $Y$ regular. Consider a dense subset $D\subset X$, a continuous map $f:D\rightarrow Y$, and a map $g:X\rightarrow Y$ (i.e. $g$ is not assumed continuous). ...
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### Continuity in a topological group

The group of the real numbers under addition is a topological group ($\mathbb{R}$,$+$) with the usual topology. However, I can't see why the group operation (addition) is a continuous function. Would ...
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### The closed points of a scheme and irreducibility

Suppose $X$ is a scheme. We say a point $x$ is closed in $X$ if $\overline{\{x\}}=\{x\}$. Let $t(X)$ be the subspace of all closed points in $X$. We say X is irreducible if its topological space is ...
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### Given a function $f$ infinitely differentiable at a point $c$ does there exist a neighborhood of $c$ in which $f$ is infinitely differentiable?

Suppose $f$ is a real valued function defined on a subset of the reals and $f$ is infinitely differentiable at $c$.Then is it possible that there does not exist any neighborhood of $c$ in which $f$ is ...
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### Locally compact space versus locally connected space

I remember when I was studying the notion of local connectedness it meant that each point has "arbitrarily small" neighborhoods that are connected. More precisely, one has the following definition: ...
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### Is SO(n) a deformation retract of SO(n+1) for $n\ge 3$?

I know that $SO(n)$ shares many topological invariants with $SO(n+1)$ for $n\ge 3$ (such as first fundamental group, first singular cohomology class over $\mathbb{Z}$, etc.). I also know that there is ...
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### Continuous functions on $J\times J\rightarrow \mathbb{R}$ can be uniformly approached by continuous functions: $f_1(x) g_1(y)+…+f_n(x) g_n(y)$

Let $J\subseteq \mathbb{R}$ be a compact interval and let $\mathbb{A}$ be a collection of continuous functions on $J \rightarrow \mathbb{R}$ which satisfy the properties of the Stone-Weierstrass ...
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### topology - stability under intersection

I am facing the following definition of topology: A topology on a set $X$ is defined as a subset $\cal O$ of the power set $\cal P$ of $X$, that: contains the empty set and $X$ is stable over the ...
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### Have I shown that convex subsets are connected with correct generality? Any flaws in my proof? (Baby Rudin 2.21(C))

$\exists a_1, b \in C_1 \subseteq R^k$ such that there is another pair of points $a_2, b \in C_2 \subseteq R^k$ where $C_1$ and $C_2$ are convex. A convex subset $C$ is defined pairwise by the ...
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### Distribution of areas of random slices through a cube

As an applied mineralogist I am always dealing with the stereology of taking measurements on a 2D slice through aggregates of multiple 3D objects, i.e. rock. If I assume the 3D object is a sphere, the ...
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### Boundedness of $f^{-1}(B)$ for some bounded set $B\in X$

If $f$ is a continuous mapping from a topological space $X$ onto itself, then is there any example of $f$ such that $f^{-1}(B)$ is not bounded for a bounded set $B\in X$?
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### Why $\{(x,y)\in\Bbb{R}^2 :\, y=x\sin \frac1x\}\cup\{(0,0)\}$ is connected but not compact?

I wonder why the set of all points in the plane satisfying $y = x\sin \frac {1}{ x}$ together with the origin is connected but not compact. Is there any example of a open cover that is not finite?
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### A simply question on open set.

Let $\mathbb{R}^n$ with the usual topology. We suppose that $A, B\subseteq \mathbb{R}^n$ are open, and suppose we have shown that $$A=B\cap C,$$ where $C\subseteq\mathbb{R}^n$ Question. Can I ...
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### Topology: Homeomorphism from R^2 to itself preserving standard topology

I need to prove that there exists a homomorphism f from R2 to itself with its standard topology such that: when considering any pair of three distinct points (x1, x2, x3) and (y1, y2, y3), f maps (...
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### Topology: What Defines (non-Trivial) Paths as being the Same Trace (Curve)?

(In the absence of feedback on this site I've also posted the question at https://mathoverflow.net/q/343762) I think I know the answer but can't prove it. Assume that Y is a Hausdorff space and ...
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### Question about extension of homeomorphism

Suppose $A$ is homeomorphic to a 2-sphere in 3-dimensional Euclidean space. Since $A$ is a closed surface in 3-dimensional Euclidean space, I wonder if the interior of $A$ together with $A$ itself can ...
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### Find out given space is $T_{2}$ or not

Question : On usual topology $R^{1}$, $\sim$ is a equivalence relation if $x-y$ is in $\mathbb{Q}$ (the set of rational numbers), then $x\sim y$. If Quotient space $R^{1}/\sim$ has quotient topology, ...
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### Is the set of convex bodies include in a closed ball compact?

I consider the set $\mathcal{K}_B$ of convex bodies (convex and compact) which are include inside the unit closed ball of $\mathbb{R}^d$. I endow this set with the Hausdorff distance. Is it compact?
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### Baire's category Theorem examples [on hold]

What are the examples of use of Baire's Category Theorem in general topology?
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### Let X=C with the usual metric and A={(x,y):y=sin1x,0<x≤1}. Show that cl(A)=A∪{(0,y):−1≤y≤1}.

Let X=$\mathbb{C}$ with the usual metric and A=$\{(x,y):y=sin\frac{1}{x},0< x \leq 1\}$. Show that cl(A)=$A∪\{(0,y):−1 \leq y \leq 1\}$. Hint: Each open ball centred at (0,y), $-1 \leq y \leq 1$ ...
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### pullback diagram & connected, irreducible or reduced schemes

Let $X, Y,Z$ schemes with maps $f:X \to Z, g:Y \to Z$. we take a look at the 'pullback' diagram $\require{AMScd}$ \begin{CD} Y \times_Z X @>p_X>> X\\ @Vp_YVV @VVfV\\ Y @>g >> Z \end{...
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### Quotient Topology and Identifying Points

I am working on some examples where we identify points and figure out what the quotient topology looks like. Below is the four examples with my guesses. Am I right? What do I need to change? The disk ...
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### $A = \{(x, y): y = \sin \frac1x , 0 < x \leq 1\}$. Show that $\operatorname{cl}(A) = A \cup \{(0, y) : -1 \leq y \leq 1\}$.

Let $X = \Bbb{C}$ with the usual metric and $A = \{(x, y): y = \sin \frac1x , 0 < x \leq 1\}$. Show that $\operatorname{cl}(A) = A \cup \{(0, y) : -1 \leq y \leq 1\}$. $\operatorname{cl}(A)$ ...
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### Reference request. Invariance of the ball by an application.

I'm writing an article and didn't want to reproduce the proof of lemma below. I just wanted to state and indicate a reference in English language to the proof of the lemma. I have a reference in ...
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### Comparison between $\sup W$ and $\sup \overline{W}$

It is well-known that $\sup S= \sup \overline{S}$ for any nonempty $S\subset \mathbb{R}$. Now, let $W$ be a nonempty subset of $\mathbb{C}^d$ and $\overline{W}$ be the closure of $W$ with respect to ...
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### A problem concerning a set’s closed intersections implying closedness

I have a feeling I didn’t quite get this problem from Munkres right. I’d like to proceed by contradiction, but feel like I’m missing some insight, which means I might be lying accidentally somewhere ...
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### Intuition for Freudenthal Suspension

One version of the Freudenthal suspension theorem is the following: Suppose a CW complex $X$ is a union of two subcomplexes $A,B$ with $A\cap B\neq\emptyset$ connected and nonempty. If $(A,A\cap B)$...
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### What does it mean to say that the space of convex bodies is locally compact with respect to the Hausdorff metric?

I know that this is equivalent to requiring each convex body in the space having a compact neighborhood. Still I can't visualize this.
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### Surface with fundamental group $\pi_1(A)\cong \mathbb{Z}_2$ embeddable in $\mathbb {R}^3$?

It’s well known that the surface formed by gluing up opposite boundary points on a closed disc has fundamental group $\mathbb{Z_2}$. However, it seems that surface can’t be realized in $\mathbb{R}^3$ ...
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### Variant Of Inscribed Square Problem

Looking at the inscribed square problem, I noticed that for some continuous curves $c:S^1\to\mathbf{R}^2$, one can prove that every continuous curve $c'$ with $\|c-c'\|_\infty<\epsilon$ contains an ...
Suppose $X$ is a metric space which is connected and has no isolated points. Then I want to show that it will have some non-empty compact subspace $Y$ such that $Y$ has no isolated points. I don't ...