Questions tagged [separation-axioms]
Separation axioms are properties of topological space which, roughly speaking, say in what way two points, a point and a closed set, or two closed sets can be "separated". Most important are $T_0$-spaces, $T_1$-spaces, Hausdorff, regular, completely regular and normal spaces.
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Is this relation making a space Hausdorff?
Given a topological space $X$ there are ways of introducing relations $\rho$ on it so that $X/\rho$ becomes a Hausdorff space. Here I will describe two relations which I will call inner and outer (...
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For compact Hausdorff spaces, is countable pseudocharacter equivalent to first countable? [duplicate]
Let $X$ be a compact $T_2$ space. Is $X$ first countable if, and only if, $X$ has countable pseudocharacter?
Note: I have already proven that every $T_1$ first countable space has countable ...
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Well-definedness of the projection associated to the sheaf of germs of a presheaf
I'm currently reading Izu Vaisman's Cohomology and differential forms ($1973$) having never studied sheaf theory before, so I will briefly write down the definitions in case they don't match with ...
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In a regular space with a $\sigma$-locally finite network, is every closed set a $G_\delta$?
In a regular space with a $\sigma$-locally finite network, is every closed set a $G_\delta$?
I'm inclined to think so based upon the basic argument of Theorem 2.8 here, but there are a few moving ...
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Understanding the Separation theorem
Separation theorem:
Let $P, Q⊆\mathbb{R}^d$ be disjoint compact convex sets. Then there exist $v∈ \mathbb{R}^d$ and $c_1, c_2∈\mathbb{R}$ with $c_1<c_2$ such that
$x.v≤c_1$ for every $x∈P$
$x.v≥...
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Character of dense subspace of regular space
Define the character of a topological space $X$ at $x$ to be the minimal cardinality of a base at $x$, denoted by $\chi(x,X)$.
Suppose $M$ is a dense subspace of a regular space $X$, prove that $\chi(...
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Must locally compact and weakly Hausdorff spaces be regular?
In a recent pull request to the π-Base, it was observed that all locally compact and KC (Kompacts are Closed) spaces are regular: since compacts are closed, each local neighborhood base of compacts is ...
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How to prove this "cofinite topology with a distinguished point" is T3.5?
In Ryszard Engelking's General Topology, its Example 1.1.8 defines a topology as follows:
Let $X$ be an arbitrary infinite set, $x_0$ a point in $X$ and $\mathcal{O}$ the family consisting of all ...
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Let $(Y,T_{y})$ be a T1 and first countable and $(X,T_{x})$ as below , prove that a function between topological spaces X to Y is continuous iff
Sorry if its elementary, but here goes.
Let X be uncountable and $x_{0}$ be an arbitrary point of X, $T_{x}$ is the topology generated by the following basis
$\beta $ = {${x}$ such that $x \neq x_0$} $...
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Cannot understand how this topology is non Hausdorff
Consider the following topology over $\mathbb{R}$: $\tau = \{\emptyset, \mathbb{R}, \mathbb{R}\setminus [0, 3], [0, 3]\}.$
I cannot understand why this is non Hausdorff. I read here around that a ...
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Two sets having strongly seperated points are strongly seperated themselves
Definition :- Two sets $A$ and $B$ are strongly seperated if there exist open neighbourhoods $U$ and $V$ such that $A \subseteq U , B \subseteq V , U\cap V=\phi$
Question :$A$ and $B$ are two compact ...
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If $(X,T)$ is a compact Haussdorff space and X is countable, then the set $\{x:\{x\} \in T\} $ is dense in $(X,T)$
If $(X,T)$ is a compact Haussdorff space and X is countable, then the set $\{x:\{x\} \in T\} $ is dense in $(X,T)$ ?
Apologize if it is elementary. but here goes what I`ve tried so far
Since that it ...
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Are Hausdorff countably compact topological groups always normal?
A colleague and I have a result that shows that for Hausdorff countably compact W-spaces, being a topological group implies normality. But it occurred to us that (being not super experienced working ...
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Singletons in regular spaces
I need help with a statement.
Let $X_1$ and $X_2$ be topological spaces and consider $X = X_1 \times X_2$. Suppose the product space $X$ is regular, that is, for any closed subset $C \subset X$ and ...
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If $X$ is regular and $A$ is a closed subset of $A$, then $X/A$ is Hausdorff [duplicate]
If $X$ is regular and $A$ is a closed subset of $A$, then $X/A$ is Hausdorff
How can I use to canonical quotient map $q: X \to X/A$ to prove the result? I tried picking distinct elements $x, y \in X/...
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