Topological spaces

Let $(X, \tau )$ be a topological space, and $\sigma$ the collection of all its closed sets. Prove that

1. $\varnothing , X \in \sigma$;

2. the intersection of an arbitrary family of sets from $\sigma$ belongs to $\sigma$;

3. the union of any finite family of sets from $\sigma$ belongs to $\sigma$. Suppose, conversely, that $X$ is some set, and $\sigma$ a family of its subsets satisfying conditions (1), (2), (3). Prove that the collection of subsets $\tau$, consisting of the complements of sets from $\sigma$, is a topology on $X$.

Let $X$ be a set, and $\beta_{0}$ an arbitrary family of its subsets forming a cover of $X$. Prove that there exists a unique topology on $X$ for which $\beta_{0}$ serves as a subbase.

Let $\left\{ \tau_{\alpha }\right\}$ be some collection of topologies on the set $X$. Prove that the system $\tau = \bigcap_{\alpha } \tau_{\alpha }$ is a topology ($\tau$ is weaker than any of the topologies $\tau_{\alpha }$).

Prove that a system of open subsets $\beta$ of a topological space $X$ forms a base of the topology $\tau$ if and only if every element of $\tau$ is a union of some elements of $\beta$.

Let $\tau_{1}$ and $\tau_{2}$ be two topologies on a set $X$ with bases $\beta_{1}$ and $\beta_{2}$ respectively. Prove that $\tau_{1}$ is weaker than $\tau_{2}$ if and only if for every $B_{1} \in \beta_{1}$ and every point $x \in B_{1}$ there exists $B_{2} \in \beta_{2}$ such that $x \in B_{2} \subset B_{1}$.

Give an example of a topological space not satisfying axiom $T_{1}$.

Prove that axiom $T_{1}$ is equivalent to the following axiom: any set consisting of a finite number of points is closed.

Give an example of a topological space satisfying axiom $T_{1}$ but not satisfying the Hausdorff axiom.

Give an example of a Hausdorff but not regular topological space.

Prove that every metric space is normal (in the metric topology).

Give an example of a regular but not normal topological space.

Prove that every metric space satisfies the first axiom of countability.

Prove that in a topological space satisfying the first axiom of countability, a point $x$ is a limit point of a set $A$ if and only if there exists a sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ of points of $A$ converging to $x$.

Prove that a metric space satisfies the second axiom of countability if and only if it is separable. Prove that a topological space with a countable base is separable, but not every separable space has a countable base.

(E. Lindelöf's property). Let $X$ be a topological space satisfying the second axiom of countability. Prove that from any collection of open sets in $X$ one can select an at most countable subcollection with the same union.

Prove that in a Hausdorff space every sequence has at most one limit. Give examples of non-Hausdorff spaces, in one of which this property nevertheless holds, and in the other it does not.

On the set of functions continuous on the segment $[0,1]$, define a topology as follows. Call a set $U$ open if every point $x \in U$ belongs to $U$ together with a neighborhood of it

where $0 \leqslant t_{1} < \ldots < t_{n} \leqslant 1$ is some collection of points, $n \in \mathbb {N}$, and $\varepsilon_{k} > 0$, $1 \leqslant k \leqslant n$. Verify the axioms of a topology and prove that this topological space is Hausdorff. Prove that a sequence $x_{n} \rightarrow x$ in this topology if and only if $x_{n}$ converges to $x$ pointwise, i.e., $x_{n}(t) \rightarrow x(t)$ for every $t \in [0,1]$. Give an example of a set and a limit point of it that is not the limit of any sequence of points of this set.

Prove that the system $\tau = \left\{ \varnothing \right\} \cap \left\{ U \subset [0,1]:[0,1] \backslash U\text{ contains at most a countable number of points}\right\}$ forms a topology. Give an example of a set $A \subset [0,1]$ and a limit point of it that is not the limit of any sequence of points of $A$.

Give an example of two incomparable ¹ topologies on one and the same set that define the same convergence.

Give an example of a separable topological space possessing a non-separable subspace.

Prove that in the space of Lebesgue-measurable functions on the segment $[0,1]$, convergence almost everywhere is not given by any topology.

Prove that in topological spaces a map is continuous if and only if the preimage of every open set is open.

Prove that in topological spaces a map is continuous if and only if the preimage of every closed set is closed.

Prove that in a topological space $(X, \tau )$ with the first axiom of countability, continuity of a function $f: X \rightarrow \mathbb {R}$ is equivalent to its sequential continuity. Give an example of a sequentially continuous but discontinuous map of a topological space.

Let $X$ be a metric space, $M_{1}, M_{2}$ closed sets in it, with $M_{1} \cap M_{2} = \varnothing$. Construct a continuous function $f: X \rightarrow [0,1]$ such that $\left.f\right|_{M_{1}} = 0$ and $\left.f\right|_{M_{2}} = 1$.

Let $X$ be a complete metric space, $M$ a closed set in it, $U_{1}, U_{2}, \ldots , U_{n}$ a cover of $M$ by open sets. Construct a system of continuous functions

called a partition of unity corresponding to the cover $U_{j}$, satisfying the conditions

1. $0 \leqslant e_{j}(x) \leqslant 1$ for every $x \in X$;

2. $e_{j}(x) = 0$, if $x \notin U_{j}$;

3. $\sum_{j = 1}^{n} e_{j}(x) = 1$ for every $x \in M$.

Let $(X, \tau )$ be a topological space. Prove that it is compact if and only if every centered ¹ system of closed sets has nonempty intersection.

Let $X, Y$ be topological spaces, and let the set $M \subset X$ be compact. Prove that:

Prove that a compact Hausdorff space (i.e., a compactum) is normal.

Let $M$ be a compact set in a topological space $X$, and $f: M \rightarrow Y$ a continuous bijective map of $M$ onto a Hausdorff space $Y$. Prove that the inverse map $f^{-1}$ is also continuous, i.e., $f$ is a homeomorphism of М and $Y$.

Prove Theorem 4.1: a subset of a metric space $X$ is compact if and only if it is closed and sequentially compact (that is, every sequence of its elements contains a subsequence converging in $X$).