# 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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### Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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### Norms and pointwise convergence

It is known that There is no norm $\| \cdot\|$ on the space $E$ of continuous real-valued functions on an interval, say $[0,1]$ such that $f_n \to f$ for $\|\cdot\|$ if and only if $f_n$ converges ...
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### How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
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### Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
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### defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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### a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
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### Is there such a thing as 'overtification' (dual to compactification)?

The dual to the notion of compactness for spaces is overtness. This duality is not manifest in the category of spaces but rather in the quantifiers used to define these notions. Is there a process ...
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### Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
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### Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
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### Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
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### When is $\{ x | f(x) \le 0\}$ path-connected?

I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x | f(x) \le 0\}$ is path-connected. We can assume that $f$ is continuous and concave (i.e. for ...
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### Topology on power set such that union is continuous

Let $C$ be a set and let $2^C$ be its power set. Consider the union function $\cup: 2^C\times 2^C\rightarrow 2^C$ such that $\cup (X,Y)=X\cup Y$. For a fixed $Y\in 2^C$, let $\cup_Y$ be the evaluation ...
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Let $\mathcal E = \{A \in M_n(\mathbb R): \rho(A) < 1\}$ where $\rho(\cdot)$ is the spectral radius and $\mathcal U$ be an affine space in $M_n(\mathbb R)$. If we assume \mathcal E \cap \mathcal U ... 1answer 224 views ### Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms? The set of Schur stable matrices is \begin{align*} \mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\}, \end{align*} where\rho(\cdot)$denotes the spectral radius of a matrix and the set of ... 0answers 116 views ### What is$\ \overline{\bigcup_{p≥ 1}\ \{A\in M_n(\mathbb C), \ A^p = I_n\}} \ $? Let$\Gamma_p = \{A\in M_n(\mathbb C), A^p = I_n\}$and let$\Gamma = \bigcup_{p≥ 1}\ \Gamma_p$. What is the closure of$\Gamma$? (This is from an oral exam). Let$B \in M_n(\mathbb C)$such that ... 0answers 1k views ###$n$-dimensional holes I am confused by the terminology concerning$n$-dimensional holes in algebraic topology. A circle is said to have a one-dimensional hole, and a sphere a two-dimensional hole for example. However I ... 0answers 195 views ### Covering a set$A \subset \Bbb R$by two families of disjoint intervals taken from given intervals Let$A \subset \Bbb R$be a bounded set. Every element$a \in A$is the center of some given open interval, let's denote it by$I_a=(a-r_a, a+r_a)$. I'm interested in knowing the following: Can we ... 0answers 422 views ### Non empty set with zero diameter Let$A \subset X$where$X$is a metric space. by definition diam$(A) = \sup\{ d(x,y), x,y \in A\}$. if$A$is non empty and has zero diameter, can I conclude that$A$is a singleton? i reason as ... 0answers 341 views ### Simpler version of dogbone space construction In "The cartesian product of a certain nonmanifold and a line is$E^4$" (R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412) Bing constructs a nonmanifold,$B$, such that$B\times \Bbb ...
A good open cover of a topological space is an open cover such that all open sets in the cover, and all finite intersections of open sets in the cover are contractible. For example, $S^2$ has an ...
### An open and connected subset $U\subseteq \mathbb C$ is still connected if you remove a curve that lies entirely in $U$
Let $U\subseteq \mathbb C$ be open and connected. If $f\colon [0,1]\to U$ is continuous with $f(0)\neq f(1)$ and $f(s)\neq f(t)$ for $s\neq t$, then $U\setminus f([0,1])$ is connected. This seems to ...