A question here on perfectly normal spaces got me into investigating the definition of such a space. The definition on wikipedia says A perfectly normal space is a topological space X in which every two disjoint non-empty closed sets $E$ and $F$ can be precisely separated by a continuous function $f$ from $X$ to the […]

Let $X$ be a compact Hausdorff space and suppose the diagonal in $X\times X$ is the zero set of a nonnegative continuous real-valued function. Why does this imply $X$ is metrizable?

My proof: Let $X$ be an space with the order topology, $x \in X$ and $F$ a closed set that does not contain $x$. Then, the set $X-F$ is an open set that contains $x$, hence there is an open set (basic) $(a,b)$ such that $x \in (a,b)\subseteq X-F$. Then $(a,b)$ and $(-\infty,a) \cup (b,\infty)$ […]

A topological space $\langle X,\tau\rangle$ is said to be normal if any two disjoint closed subsets are separated by open sets, meaning that for disjoint $E,F\subseteq X$ with $X\setminus E,X\setminus F\in\tau,$ there are disjoint sets $U,V\in\tau$ such that $E\subseteq U$ and $F\subseteq V.$ We say that $A,B\subseteq X$ are separated by a continuous function if […]

I have a simplicial complex $K$ and I need to show that its topological realisation $|K|$ is Hausdorff. And $K$ need not be finite. I have very little idea on how to get started on this. Only that if $x,y \in |K|$, I need to find disjoint open sets containing $x$ and $y$. I also […]

Show that a locally compact Hausdorff space $(X,\tau)$ is regular. I have already shown that a compact Hausdorff space is regular. My textbook proposes 2 methodes, but I get stuck at both. The first method looks the most elegant, but how can I continue? Compactification method. Consider the compactification $(X_\infty, \tau_\infty)$ where $X_\infty = X\sqcup […]

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am namely talking about the $T_{k}$ and $N_{k}$ hierarchy. I understand the Hausdorff property, i.e. $T_{2}$-property, is important in analysis and limits; and […]

I have some problems understanding the proof of the following lemma: Lemma: Let $x \in X, \ \ \ U, W \in \mathcal{U}, \ \ \ \mathcal{T(U)}$ is the topology on $X$. If there exists $V \in \mathcal{U}$ such that $U \circ V \subset W$, then $U[x] \subset intW[x]$ and $\overline{U[x]} \subset W[x]$. $\mathcal{U}$ is […]

Are there any simple examples of subspaces of a normal space which are not normal? I know closed subspace of a normal space is normal, but open subspace in most cases which I can think of are also normal.

Let $(X,\tau)$ be a $T_1$-space and $X$ is an infinite set. Then $(X,\tau)$ has a subspace homeomorphic to $(\mathbb{N},\tau_2)$, where $\tau_2$ is either the finite-closed topology or the discrete topology. Update attempt: As suggested from Daniel fischer comment. First i am going to prove a simple lemma to begin with. Lemma : If $(X,\tau)$ is […]

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