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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.

57719 questions
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Connected topology on $\mathbb Q$ such that every point is a cut point?

Is there a $\text{T}_1$ topology on $\mathbb Q$ with these properties? It should be connected such removing any point disconnects the space.
Forever Mozart
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Prove that something is an embedding, with and without using formulas

Show that, for any $n\geq1$ integer, $$S^1\times...\times S^1\subset\mathbb{R}^2\times...\times\mathbb{R}^2=\mathbb{R}^{2n}$$ can be embedded in $\mathbb{R}^{n+1}$. I first have to prove it without using explicit formulas, then by explicit…
jbuser430
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hausdorff, intersection of all closed sets

Can you please help me with this question? Let's $X$ be a topological space. Show that these two following conditions are equivalent : $X$ is hausdorff for all $x\in X$ intersection of all closed sets containing the neighborhoods of $x$ it's…
Lilly
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Point set topology question: compact Hausdorff topologies

$\tau_1,\tau_2,\tau_3$ are topologies on a set such that $\tau_1\subset \tau_2\subset \tau_3$ and $(X,\tau_2)$ is a compact Hausdorff space. Could any one tell me which of the following are correct? $\tau_1=\tau_2$ if $(X,\tau_1)$ is compact…
Myshkin
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Closure of intersection of two sets

I have a problem with the following exercise: Let be $A,B$ subsets of a topological space $X$. Prove that $$\overline{A\cap B}\subset \bar{A}\cap\bar{B}.$$ I only know is that $\bar{A}=A\cup\partial A$.
MathCracky
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Homeomorphism between $\mathbb R^n\setminus\{0\}$ and $S^{n-1}\times \mathbb R$

I am trying to find a homeomorphism between $\mathbb{R}^n\backslash\{0\}$ and $\mathbb{S}^{n-1}\times\mathbb{R}$. I read some related questions and am thinking of the map $f:\mathbb{R}^n\backslash\{0\}\rightarrow\mathbb{S}^{n-1}\times\mathbb{R}^+$…
user194469
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Why does the notion of vanishing at infinity require local compactness|

The definition I know of 'vanishing at infinity' for locally compact topological space is the following: A function $f:X\to (Y,||\cdot||)$ on a locally compact space $X$ is said to vanish at infinity if for every $\epsilon>0$ the set $\{x\in X|\…
user2520938
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Dealing with connectness and compactness of matrices.

Consider the set of all $n\times n$ matrices with real entries, considered as the space $\mathbb{R^{n^2}}$ What can we say about connectedness and compactness of the following sets? The set of all orthogonal matrices. The set of all matrices with…
Srijan
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The approximation of set

$A,B$ are closed sets of $\mathbb R^n$ such that $A \subset {\rm{Int}}(B)$. Is there a positive continuous function $\sigma (x)$ on $B$ such that if $f$ is a continuous function from $B$ to $\mathbb R^n$ such that $\left| {f(x) - x} \right| < \sigma…
Summer
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Is $(U \cap A \neq \emptyset) \Leftrightarrow (U \cap \overline{A} \neq \emptyset)$ only true in Hausdorff spaces?

$A$ is a subset of a topological space $X$, in which $U$ is open. I'm asking because I was looking at these exercises (this is the last one), and it specifies that $X$ is Hausdorff. Here's my attempt of a proof: '$\Rightarrow$': $\emptyset…
man_in_green_shirt
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Topological invariants

Do continuous maps necessarily preserve topological invariants? Or is it necessary for the maps to be homeomorphisms? Are there simple examples where continuous maps do not preserve these invariants?
freddie
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A problem about connected spaces

Possible Duplicate: Bourbaki exercise on connected sets Let $X \subset M$ be both connected spaces. If $C$ is a connected component of $M-X$, then $M-C$ is connected. My try: Let´s suppose that $A,B$ form a separation of $M-C$ i.e $ M - C = A…
Arkj
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Show that a star-like region is simply connected.

A set S is called star-like if there exists a point $\alpha\in S$ such that the line segment connecting $\alpha$ and z is contained in S for all $z\in S$. Show that a star-like region is simply connected. My answer Show that $γ:γ(t)=tz+(1−t)α, t≥1$…
Breton
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Which is the domain set?

Let $X \subseteq \mathbb{R}$ and $f,g : X \rightarrow X $ be continuous functions such that $f(X) \cap g(X) = \emptyset$ and $f(X) \cup g(X) = X$. Then which of the following cannot be $X$ ? A. $[0,1]$ B. $(0,1)$ C. $[0,1)$ D.…
user118494
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Prove that if Diagonal is open in Product Topology, then the original topology is discrete

I found the following exercise in Introduction to Metric and Topological Spaces by Sutherland (Chapter 10 Question 20). Prove that the topology on a space X is discrete iff the diagonal $\Delta=\{ (x,x) \mid x\in X\}$ is open in the topological…
BeerR
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