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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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If $A$ and $B$ are two disjoint non empty subsets of $\mathbb{R}^2$ such that $A\cup B$ is open....

Question is : If $A$ and $B$ are two disjoint non empty subsets of $\mathbb{R}^2$ such that $A\cup B$ is open in $\mathbb{R}^2$ . Then which of the following is true? If $A$ is open and $A\cup B$ is connected then $B$ must be closed in…
user87543
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Minimal subbases for the canonical topology on $\mathbb{R}$?

Thoughts: I know that every base for the canonical topology can be reduced. So I'm guessing that maybe the canonical topology doesn't have a minimal subbase? Would it be possible to use the fact that every base for the canonical topology on…
tcmtan
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Is there a continuous function $f:S^1 \to \mathbb R$ which is one-one?

Is there a continuous function $f:S^1 \to \mathbb R$ which is one-one?
La Rias
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compactness of the real projective plane

Let $\mathbb{P}_2(\mathbb{R})$ =( $\mathbb{R}^3$-{0})/~ where x~$r$x for any nonzero point x $\in \mathbb{R}^3$ and any nonzero $r \in \mathbb{R}$ I want to show that the $\mathbb{P}_2(\mathbb{R})$ is compact.. I used quotient map to solve, but it…
NNNN
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Why does Munkres define functions in a seemingly complicated way?

In Munkres's extremely careful and well-written textbook, Topology (2nd Edition), he defines functions as follows. First he defines a rule of assignment as a subset of a cartesian product $r\subset C\times D$ where each element of $C$ appears as the…
Cheerful Parsnip
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Condition for a compact set to be the support of a continuous function?

I am studying Rudin's Real and Complex Analysis exercises and I am currently thinking about the following: Is there a characterization of the class of compact sets of $\mathbb{R}$ which are supports of continuous functions? Is this characterization…
TerranDrop
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Derived sets - prove $(A \cup B)' = A' \cup B'$

I'm trying to prove $(A \cup B)' = A' \cup B'$ where $S'$ denotes the derived set of some subset $S$ of a topological space $X$. A derived set $S'$ of a set $S$ is the set of $x \in X$ such that $x$ is in the closure of $S-\{x\}$. I'm having…
goatman2743
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What exactly are the properties of Topology?

I've looked through quite a few resources on the internet and I can't really find a good answer to this question. I am just starting to get into topology and manifolds and I came across the statement "A coffee mug and a torus are topologically the…
Spaderdabomb
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Is there a bijective continuous function $f: \Bbb R \to \Bbb R$ whose inverse $f^{-1}$ is not continuous?

Looking at the definition of an homeomorphism, this question came to my mind.
user93718
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A Non-Metrizable subspace of a compact

Let $K$ be a Hausdorff compact space in which every closed set is $G_{\delta}$. Suppose that $X$ is a non-metrizable subspace of $K$. Prove that $X$ cannot be written as $X=\bigcup_{n=1}^{\infty}X_{n}$ where each $X_{n}$ is a metrizable subspace of…
Maksim
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Confused about the definition of a limit point

I am studying topology, and the definition of a limit point on my book is: Let $A$ be a subset of a topological space $X$ and $x$ a point of $X$, we say that $x$ is a limit point of $A$ if every neighborhood of $x$ intersects $A$ at some point other…
Coco
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Soft question: Which topological properties are transferred by bijective continuous maps?

Note The following question was asked by @Tian Vlašić, who then closed it after several interesting comments had been made. I was interested enough that I wanted to revive it. Let us say that a property $P$ of topological spaces is transferred by…
John Hughes
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$F_\sigma$-subsets in a normal space can be separated

Let $X$ be a normal Hausdorff space and let $C,D$ be two $F_{\sigma}$ subsets of $X$ such that $\overline{C} \cap D = \emptyset$ and $C \cap \overline{D} = \emptyset$. Prove there exists disjoint open subsets $U,V$ such that $C \subset U$ and $D…
user10
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Whether $(0,1)$ and $(0,1]$ are homeomorphic

In connection with the question continuous onto map from $(0,1)\to (0,1]$ I would like to know whether $(0,1)$ and $(0,1]$ are homeomorphic. The map mentioned in the above question is onto but not a bijection. So does such a continuous bijection…
Sriti Mallick
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What does it mean for a topological space to be "normal" (T4)?

I'm working through Topology Through Inquiry, and while I grok the meaning of T1 and T2 spaces (they roughly quantify the ability of a topology to separate the elements in the space), what the meaning of T3 and T4 spaces is less clear. I'm not…
J C
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