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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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Every closed set in $\mathbb R^2$ is boundary of some other subset

A problem is bugging me many years after I first met it: Prove that any closed subset of $\mathbb{R}^2$ is the boundary of some set in $\mathbb{R}^2$. I have toyed with this problem several times over the last 20 years, but I have never managed to…
Old John
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Dense and locally compact subset of a Hausdorff space is open

Let $X$ be a Hausdorff space and let $D \subseteq X$ be locally compact and dense in $X$. Why is $D$ open? I can see that $D$ is regular but don't see why $D$ is in fact open.
user10
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16
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4 answers

Continuous functions on a compact set

Consider the following statement: Let $K$ be a nonempty subset of $\mathbb R^n$, where $n\geq1$. Then If $K$ is compact, then every continuous real-valued function defined on $K$ is bounded. Here are my questions: Is the converse true? (If…
user9464
16
votes
1 answer

Normal + Connected -> Uncountable

Here's a solution (to the exercise "prove that if $X$ is a space with more than one element, and is normal and connected then $X$ is uncountable"): by Urysohn's lemma, given $A$ and $B$, closed and disjoint in $X$, there exists a continuous…
Weltschmerz
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16
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2 answers

show that a distance function is continuous

Let $X$ be a metric space with metric $d$. Define $d: X \times X \to \mathbb{R}$, show that $d$ is continuous. I would like to show that the function is continuous the topology way (since it is a course on topology). Let $(a,b)$ be a basic open set…
Daniel
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16
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2 answers

Is the closure of connected set path connected?

If $U$ is a connected open set in $\mathbb R^n$, is $\bar U$ path-connected? (that is, for any two points $x_1,x_2$ in the closure of $U$, can we find a continuous path connecting them?)
Summer
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16
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4 answers

Why don't we define neighbourhoods to be open sets?

There seems to be two common definitions of neighborhoods of a point- an open set containing the point, or an set containing an open set which itself contains that point. Why don't we just use the former definition, it seems so much simpler? What do…
user6873235
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16
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4 answers

$f$ is Continuous if and only if its Graph is Closed in $X \times Y$

Let $f: X \to Y$; let $Y$ be compact Hausdorff. Then $f$ is continuous if and only if $G_f = \{(x,f(x)) \mid x \in X \}$ is closed. Here is my shot at a proof: Suppose that $f : X \to Y$ is continuous, and let $(x,y) \in \overline{G_f}$ but…
user193319
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16
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Exhaustion of open sets by closed sets

Let $X$ be a topological space with topology $\tau$. Let $U\in \tau$. Say that $U$ can be countably exhausted by closed sets if there exists a family of sets $F_n \subset U$ indexed by $n\in\mathbb{N}$ such that $F_n \subset V \subset F_{n+1}$ for…
Willie Wong
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Separation in direct limits of closed inclusions

Suppose $X$ is a space and $A_1\subseteq A_2\subseteq A_3\subseteq ...\subset X$ is a sequence of subspaces each of which is closed in $X$ and such that $X\cong \varinjlim_{n}A_n$ (i.e. $U$ is open in $X$ if and only if $U\cap A_n$ is open in $A_n$…
J.K.T.
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15
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2 answers

Lie group, differential of multiplication map

Let $G$ be a Lie group Let $m:G\times G\rightarrow G$ denote the multiplication map. Show that the differential $dm_{(e,e)}:T_eG\oplus T_eG\rightarrow T_eG$ is given by $$dm_{(e,e)}(X,Y)=X+Y$$ This question comes from Lee, introduction to smooth…
user113913
15
votes
3 answers

$\Bbb RP^2$ as the union of a Möbius band and a disc

I spent some time trying to understand that the projective plane, $\Bbb RP^2$, can be viewed as the union of a Möbius band and a disc. I consider this using Homogeneous coordinates. But I still hope to really 'see' this fact in some way.
Xiaochuan
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15
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Open Sets Boundary question

I'm really having a hard time with this problem: Find three disjoint, open sets in $\mathbb{R}$ (std. topology) that have the same nonempty boundary. I played around with a few ideas like $\mathbb{Q}$, {$\sqrt{p}+\mathbb{Q}$},…
user124910
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15
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4 answers

What does it mean to induce a topology?

I am reading the definition of "metrizability" which states that if there exists a metric $d$ on set $X$ that induces the topology of $X$, then it is metrizable. My question is how can we possibly know what topology is being induced by metric $d$…
Rutherford Mark
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The closed graph theorem in topology

Theorem A map $\phi$ maps a topological space $X$ to another $Y$, where $X$ is Hausdorff, $Y$ is compact and the graph of $\phi$ is closed. Then $\phi$ is continuous. Is it reallly necessary to include the condition that $X$ is Hausdorff? Since I…
awllower
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