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Give examples of ordered sets $(X,\leq)$, which are: linearly ordereded, inductively ordered, well-ordered.
Describe the metric topology on a metric space $(X,d)$.
Explain the notions of a topological space and of a continuous map from one topological space into another topological space.
Give a necessary and sufficient condition for a continuous mapping $f:X\rar Y$ to be closed.
Define the notions of a filter and an accumulation point of a filter.
Give an example of an l.s.c function, which is not continuous.
Explain the notions of initial and final topologies. Give examples of each of them.
What is a Hausdorff-space? Give examples of topological spaces, which are Hausdorff and which are not.
Suppose $X_\a$ are Hausdorff and $X$ carries the initial topology with respect to the mappings $f_\a:X\rar X_\a$. Give a necessary and sufficient condition for $X$ to be Hausdorff.
Define the attaching $X\cup_f Y$ of topological spaces $X$ and $Y$ along a closed subset $A$ of $X$ by a continuous mapping $f:A\rar Y$.
Why is the projective space $P^n(\R)$ Hausdorff?
Explain the notions of a regular, a completely regular and a normal space. Why is a normal space completely regular?
Suppose $X$ is normal and $R$ is a closed equivalence Relation on $X$. Explain the following result: $X/R$ is normal if the quotient map $X\rar X/R$ is either closed or open.
Give a necessary and sufficient condition for a topological space to be separable and metrizable.
Suppose $X$ is a compact metric space, $a\in X$ and $H$ the set of all $1$-Lipschitz functions $f:X\rar\R$ such that $f(a)=0$. Show that $H$ is compact in $C(X)$.
Let $f$ be an integrable none-negative function on $(\O,\F,\mu)$ and $H\colon=\{g\in L_1(\mu):|g|\leq f\}$. Show that $H$ is weakly compact in $L_1(\mu)$.
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Last modified: Tue Sep 5 10:40:22 CEST 2023