Questions tagged [tychonoff-spaces]
For questions involving Tychonoff spaces, or topological spaces satisfying the $T_{3 \frac 1 2}$ separation axiom.
15 questions
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Equivalent characterizations of Stone-Cech compactification and generalization to Wallman-Frink compactifications
In the book Normal topological spaces by Alo and Shapiro the following theorem is present:
Theorem. If $X$ is a dense subspace of Tychonoff space $Y$ then the following are equivalent:
Every ...
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How to prove that he looped line topology is Tychonoff? [closed]
I came to this question when I was solving a problem in Willard's General Topology (p. 36).
I am new to topology and I want to prove that the looped line topology is Tychonoff space. My idea was to ...
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Thomas plank is not realcompact
Let $X = \bigcup_{n\geq 0} L_n$ where $L_n = [0, 1)\times\{1/i\}$ for $i > 0$ and $L_0 = (0, 1)\times \{0\}$.
Define the topology on $X$ as follows: each point $(x, 1/i)$ for $x\in (0, 1)$ and $i &...
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Mysior plane is not realcompact
Let $X = \mathbb{R}^2$ with $(x, y)\in X$ for $y\neq 0$ isolated and $(x, 0)$ having neighbourhood basis of the form $$U_n(x) = \{(x, y) : y\in (-1/n, 1/n)\}\cup \{(x+y+1, y) : 0 < y < 1/n\}\cup ...
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Weight of the Stone-Cech compactification of a Tychonoff space
Let $w(X)$ be weight of $X$, that is least infinite cardinality of a basis of $X$.
Here $X$ is assumed to be Tychonoff.
Is the inequality $w(\beta X)\leq 2^{w(X)}$ true? This seems to hold for many ...
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4
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Maximal Non-Hausdorff Compactification
I have recently started to read about compactifications of topological spaces, however I would like to clear my mind on a few things.
For starters, I am interested in generic topological spaces (not ...
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If $f(x)$ and $f(C)$ are separated by neighborhoods in $\mathbb R$, are $x$ and $C$ separated by a function?
Let $X$ be a topological space, $C\subseteq X$ be closed, $X\ni x\notin C$, and $f\in C(X,\mathbb R)$. Suppose further that $f(x)$ and $f(C)$ are contained in disjoint open neighborhoods of $\mathbb R$...
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If $f(A)$ and $f(B)$ are separated by neighborhoods in $\mathbb R$, are $A$ and $B$ separated by a function?
Let $X$ be a topological space, $A,B\subseteq X$, and $f\in C(X,\mathbb R)$. Suppose further that $f(A)$ and $f(B)$ are contained in disjoint open neighborhoods of $\mathbb R$. Is this enough to ...
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How to prove that the Stone-Čech compactification of a Tychonoff space always exists?
I am wondering how to prove the following theorem:
Let "X" be a Tychonoff space. Then its Stone-Čech compactification
exists and it is unique (up to homeomorphism).
The uniqueness part is ...
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1
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Tychonoff spaces and separating points.
A few months ago I read here or elsewhere that a Hausdorff space X is completely regular if it satisfies an equivalent weaker separation axiom:
Every two different points x, y in X can be separated by ...
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Clarifying the definition of the Tychonoff space $T$ in Fuchs-Fomenko's book
In their topology book, Fuchs and Fomenko define the Tychonoff space $T$ to be the set of all real sequences $(x_1, x_2, x_3, ...)$ with the base of topology formed by the sets $\{(x_1, x_2, x_3, ...) ...
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Borel $\sigma$-algebra of separable $T_{3 \frac 1 2 }$ space generated by bounded continuous functions
A topological space $\Omega$ satisfies the $T_{3\frac 1 2}$ separation axiom if for every $A \subset \Omega$ closed, and every $x \in \Omega \setminus A$, there is a continuous function $f : \Omega \...
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Constructing sequence of nested closed sets in separable $T_{3\frac 1 2}$ space
Let $X$ be a separable $T_{3\frac 1 2}$ space. In other words, for every closed $A \subset X$ and every $x \in X \setminus A$, there is a continuous function $f : X \to [0,1]$ for which $f(x) = 0$ and ...
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How to best "sort" Tychonoff spaces, so it is mutually exclusive, collectively exhaustive?
How to best "sort" Tychonoff spaces, so it is mutually exclusive, collectively exhaustive?
Why am I asking: When describing certain mathematical phenomenon in a paper, it may be useful to go ...
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A topological space $X$ has a compactification if and only if $X$ is a Tychonoff space
Following a reference from "General Topology" by Ryszard Engelking
Lemma
Let be $(X,\mathcal{T})$ a not compact topological space and let be $\infty\notin X$; thus on $X^\infty=X\cup\{\infty\}$ we ...
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