Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [tychonoff-spaces]

For questions involving Tychonoff spaces, or topological spaces satisfying the $T_{3 \frac 1 2}$ separation axiom.

Tychonoff spaces are topological spaces satisfying the $T_{3 \frac 1 2}$ separation axiom: a space $X$ is Tychonoff if for every closed subset $A \subset X$, and every point $x \in X \setminus A$, there is a continuous function $f : X \to [0,1]$ for which $f(x) = 0$ and $f(y) = 1$ for every $y \in A$. In particular, every metric space is Tychonoff, and every Tychonoff space is regular and Hausdorff.

19 questions
9
votes
1 answer

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 $$U_n(x) = \{(x, y) : y\in (-1/n, 1/n)\}\cup \{(x+y+1, y) : 0 < y < 1/n\}\cup \{(x+\sqrt{2}+y, -y) : 0 < y < 1/n\}.$$ The space $X$…
Jakobian
  • 15,280
5
votes
1 answer

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 continuous function $f:X\to Z$ where $Z$ is compact…
Jakobian
  • 15,280
4
votes
1 answer

Existence of non-discrete Tychonoff extremally disconnected P-space and measurable cardinals

Assume all spaces are Tychonoff. In exercise $12$H of Rings of continuous functions by Gillman and Jerison they ask to prove the following two things: If $X$ is an extremally disconnected $P$-space of cardinality smaller than first measurable…
Jakobian
  • 15,280
4
votes
1 answer

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 $i > 0$ is isolated, neighbourhoods of $(0, 1/i)$ are…
Jakobian
  • 15,280
4
votes
1 answer

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 necessarily Tychonoff) although I imagine that not…
AlienRem
  • 4,130
  • 1
  • 27
  • 38
3
votes
2 answers

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 consider the topology $$ \mathcal{T}^\infty:= \{U…
Antonio Maria Di Mauro
  • 5,226
2
votes
2 answers

Is $\mathbb{N}$ the only discrete dense subspace of $\beta \mathbb{N}$?

This is related to my last question. Since a discrete $X$ is locally compact Hausdorff, its Stone-Čech compactification $i:X\rightarrow \beta X$ is a homeomorphism onto an open dense subspace of $\beta X.$ Thus $\beta X$ contains a dense discrete…
Miles Gould
  • 1,284
2
votes
1 answer

Which spaces admit only extremally disconnected compactifications?

A compactification is an embedding into a Hausdorff compact space. In my other question I've considered locally compact Hausdorff spaces which admit only zero-dimensional compactifications, which turn out to be precisely the zero-dimensional spaces…
Jakobian
  • 15,280
2
votes
1 answer

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 spaces. I tried looking in Engelking for an answer…
Jakobian
  • 15,280
2
votes
1 answer

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 $f : \Omega \to [0,1]$ for which $f(x) = 0$ and $f(y) = 1$ for…
D Ford
  • 4,327
1
vote
1 answer

Topological spaces with dense discrete subspaces

Let $X$ be a totally disconnected compact Hausdorff space. Does $X$ always have a discrete dense subspace $Y$? My motivation for asking this is that the Stone-Čech compactification of a discrete space is always totally disconnected. If the answer is…
Miles Gould
  • 1,284
1
vote
0 answers

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 $\mathbb R$. Is this enough to enough to ensure that $x$…
WillG
  • 7,382
1
vote
1 answer

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 enough to ensure that $A$ and $B$ are separated by a…
WillG
  • 7,382
1
vote
0 answers

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 a continuous function f on X. Is it true?
Marek Wojtowicz
  • 11
0
votes
1 answer

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 clear to me, but what seems difficult is check that $\beta…
Tereza Tizkova
  • 2,750
1
2