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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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example of a totally bounded but not bounded

I have got an question:- is there any counter-example of totally bounded but not bounded space can anyone help me please.thanks for your help
priti
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Topology on set of maps from $A$ to $B$

Let $B$ be a non-empty set equipped with the discrete topology, and let $A$ be an infinite set. Then $B^A$ is the set of all functions $f:A\to B$. I have to verify some elementary properties of the product topology that $B^A$ inherits: Prove that…
Dr. Heinz Doofenshmirtz
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If $K$ is compact, prove that $K$ has a minimum and maximum.

Let $X$ be a nonempty totally ordered set which doesn't have a minimum or maximum and let $T$ be order topology on $X$. Let $ K \subset X $ be a nonempty subset of $X$. If $K$ is compact, prove that $K$ has a minimum and maximum. I tried by…
user15269
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Closed and bounded set in $\mathbb{R}^n$ is sequentially compact

I want to prove that any closed and bounded set in $\mathbb{R}^n$ is sequentially compact. Note that I want to prove this without using the Heine-Borel theorem and the equivalence of sequential compactness and compactness. Ah and I can use that any…
elaRosca
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Alternative definition for topological spaces?

I have just started reading topology so I am a total beginner but why are topological spaces defined in terms of open sets? I find it hard and unnatural to think about them intuitively. Perhaps the reason is that I can't see them visually. Take…
Mark
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Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected

Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected. (Hint:Show that given $x_0 \in U$, the set of points can be joined to $x_0$ by a path in $U$ is both open and closed in $U$.) This should not be too…
Daniel
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Help understanding $T_1$ & $T_2$ spaces.

What are $T_1$ & $T_2$ spaces? I'm having trouble understanding what exactly are $T_1$ & $T_2$ spaces.
T1mbo
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Kuratowski closure-complement question involving unions and intersections

Suppose $X$ is a topological space containing subsets $A$ and $B$ such that $$\tag1ikA=ikiA=iA\subsetneq A=kA=kikA=kiA$$ $$\tag2kiB=ikiB=iB\subsetneq B\subsetneq kB=kikB=ikB$$ where $k$ is closure and $i$ is interior. These relations, as well as…
mathematrucker
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How to show that $f(S) \subset Y $ is dense, when $f$ is continuous and surjective, and $S \subset X$ is dense in $X$?

Let $(X,d_x)$ and $(Y,d_y)$ be metric spaces. Furthermore, let $f:X \to Y$ be surjective and continuous. Furthermore: $S \subset X$ is dense in X. Question: How to prove that $f(S) \subset Y$ is dense in Y? I wrote down the definitions of…
Max Muller
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Question on closure in the product and box topologies

Hi everyone this is my first post so apologies in advance if I do anything wrong... I'm working through an unmarked assignment sheet and am struggling on this question: Let $X = \prod^{\infty}_{i=1} X_i$ where $X_i = \mathbb{R}$ for all $i$. Let…
120 Eyes
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Give an example of a set $A$ for which the following sets are pairwise different.

Give an example of a set $A$ for which the sets: $A, \text{Int}(A), \overline A, \text{Int}(\overline A), \overline {\text{Int}(A)}, \text{Int}(\overline{\text{Int}( A)}), \overline{\text{Int} \big( \overline A\big)} $ Are pairwise different. My…
Dima
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Why, in topology, is continuity defined with inverse images?

In topology we defined a continuous map to be a map between 2 spaces $f: X \mapsto Y$ such that if $U\subset Y$ is open then $f^{-1}(U)$ is open. Why did we use the preimage; why not say 'a continuous map always maps open sets to open sets'?
Toby Peterken
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topology induced by function from set to power set?

tl;dr: I've tried to construct a different way to formalize "topological spaces" than via open sets or neighborhoods. I have not seen this approach but it may have been done before. The definition as it currently stands is not satisfactory (see…
user56834
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Reverse of Jordan curve theorem

Let $K$ be a compact subset of $\mathbb R^2$ such that $\mathbb R^2\setminus K$ is not connected. Is it true that $K$ contains a simple closed curve?
Emanuele Paolini
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Nested homeomorphic sets

Suppose we have a countable collection of sets $\{U_n\}$ such that $U_n\subset U_{n+1}$ for each $n$ and $U_n$ is homeomorphic to $\mathbb{R}$ (or more generally, $X$) for each $n$, then is $\bigcup_{n=1}^\infty U_n$ homeomorphic to $\mathbb{R}$ (or…
J.Doe
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