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Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

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Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$[NBHM_PhD Screening Test-2013, Topology]

Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$ (a) The set of all upper triangular matrices all of whose eigenvalues satisfy $|\lambda|≤2 $. (b) The set of all real symmetric matrices all of whose eigenvalues satisfy…
poton
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Every retraction is a quotient map?

I have to proof that every retraction is a quotient map.. I have no idea where to start or what to use! A retraction $r:X \rightarrow A$ is a continuous map s.t. $r(a)=a$ for every $a\in A$.
user57012
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Prove a closed set is a countable union of compact sets, in $\mathbb{R}^n$.

I'm studying Sard's theorem and I want to know why is true that, in $\mathbb{R}^n$, every closed subset can be expressed as a countable union of compact sets. Thank you, :)
Alexandre Muñoz
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Generalize continuity from compact subsets to the whole domain space

I saw a statement on http://en.wikipedia.org/wiki/Metric_space#Continuous_maps: f is continuous if and only if it is continuous on every compact subset of M1. It is stated for the case that f is a mapping between two metric spaces. I was wondering…
Tim
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Why do we care about quotient maps?

In topology class, I often hear statements like "It turns out that $f$ is always a quotient map." I don't understand what is so special about quotients that the question of whether a given surjective map is or isn't a quotient is so interesting. …
lily
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Let $(\mathbb{N},\tau)$ be a topological space, where $\tau=\{\emptyset, \mathbb{N}, \{0\},\{0,1\},\{0,1,2\},\dots\}$

I'm stuck here: Let $(\mathbb{N},\tau)$ be a topological space, where $$\tau=\{\emptyset, \mathbb{N}, \{0\},\{0,1\},\{0,1,2\},\dots\}$$ a) Prove that is not compact. b) Prove that every continuous function $f: (\mathbb{N},\tau) \to \mathbb{R}$ is…
Alure
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Is countable intersection of perfect set perfect? And one question about closed connected set

Let $\{A_n\}$ be a sequence of perfect sets in a metric space. Then, is $\bigcap_{n\in\omega} A_n$ perfect? Recently, i have studied some Cantor-like sets and i got this very natural question. Let $C_1$ be a closed connected set in a metric space…
Katlus
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Is the union of a chain of topologies a topology?

Let's consider the following inclusion chain of topologies on space $X$: $\tau_1\subset\tau_2\subset\cdots\subset\tau_n\subset\cdots$. Let $\tau=\bigcup_{n=1}^\infty \tau_n$. Is $\tau$ a topology? Obviously , the intersection of any two sets from…
pabodu
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Help defining a topological property

After an introductory course in General Topology I started to look up many examples of topologies (mainly on $\mathbb{R}$) just to get a feel for how different they can be and it seems to me that they all can be fit in one of the following…
Ettore
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Zero set in zero set need not be zero set

A subset $A$ of a topological space $X$ is called a zero-set if $A=f^{-1}(0)$ for some continous function $f:X\to\mathbb{R}$ or equivalently for some continuous $f:X\to[0,1]$. What would be an example of a zero-set $M$ in a space $X$ such that there…
PatrickR
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When are all subsets intersections of countably many open sets?

(Motivated by a comment by Jeppe Stig Nelson to this question) In what topological spaces $X$ can every subset $A$ be written as the intersection of countably many open sets, i.e., $A=\bigcap_{n=1}^\infty U_n$ with $U_n$ open? Without demanding…
Hagen von Eitzen
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property about topological space

I have a question. Let $f,g$ be continuous functions from $X$ to $Y$, $X$ is a topological space and $Y$ a topological space under ordered topology. Then prove that the set $\{x \in X \ | \ f(x) < g(x)\}$ is open. I want to know that what intrisic…
M.Subramani
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If a topological space can be partitioned into finitely many second countable subspaces, is it second countable? [Collecting examples]

A topological space is second countable if its topology has a countable basis. Let $(X,\tau)$ be a topological space. Suppose there exists $S_1,\ldots, S_n$ is a finite partition of $X$ such that, for each $1\leq i \leq n$, the subspace…
Anguepa
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Pseudo metric spaces are not Hausdorff.

I know that every metric space is Hausdorff,and every metric space is Pseudo metric. How can I prove that every Pseudo metric space is not Hausdorff??
ccc
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